WAVELENGTH TUNABLE DEVICES BASED ON HOLOGRAPHIC POLYMER DISPERSED LIQUID CRYSTALS. A dissertation Submitted to. Kent State University

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1 WAVELENGTH TUNABLE DEVICES BASED ON HOLOGRAPHIC POLYMER DISPERSED LIQUID CRYSTALS A disstatio Submittd to Kt Stat Uivsity i patial fulfillmt of th quimts fo th dg of DOCTOR OF PHILOSOPHY By Hailiag Zhag May, 8

2 Disstatio witt by Hailiag Zhag B.S., Xiagta Uivsity, 987 M.S., Kt Stat Uivsity, 999 Ph.D., Kt Stat Uivsity, 8 Appovd by, Jac R. Klly, Pofsso, Chai, Doctoal Disstatio Committ, Ggoy P. Cawfod, Pofsso, Mmbs, Doctoal Disstatio Committ, Dg-K Yag, Pofsso, Mmbs, Doctoal Disstatio Committ, Eug C. Gatlad, J, Pofsso, Mmbs, Doctoal Disstatio Committ, Qi-Huo Wi, Assistat Pofsso, Mmbs, Doctoal Disstatio Committ, Doald L. Whit, Pofsso, Mmbs, Doctoal Disstatio Committ Accptd by Olg D. Lavtovich Timothy S. Molad, Chai, Liquid Cystal Istitut, Da, Collg of Ats ad Scics ii

3 ZHANG, HAILIANG, Ph.D, May 8 CHEMICAL PHYSICS WAVELENGTH TUNABLE DEVICES BASED ON HOLOGRAPHIC POLYMER DISPERSED LIQUID CRYSTALS, () Dicto of Disstatio: Jac Klly Wavlgth tuabl dvics hav gatd gat itst i basic scic, applid physics, ad tchology ad hav foud applicatios i Lida dtctio, spctal imagig ad optical tlcommuicatio. This thsis focuss o th physics, tchology ad applicatio of sval wavlgth tuabl dvics basd o liquid cystal tchology, spcially o Hologaphic Polym Dispsd Liquid Cystals (HPDLC). HPDLCs a fomd though th photo-iducd polymizatio pocss of photopolymizabl mooms, ad slf-diffusio pocss ad phas spaatio pocss of th mixtu of liquid cystals ad mooms, wh th mixtus of liquid cystals ad mooms a xposd to th itfig moochomatic light bams. Th ifomatio fom th itfig patt is codd ito th hologaphic liquid cystal/polym composits, which a switchabl o tuabl upo xtal lctic filds. Basd o th lctically cotollabl bam stig capability of tasmissio HPDLCs, ovl switchabl cicula to poit covt (SCPC) dvics a dmostatd fo slctig ad outig th wavlgth chals discimiatd by a Faby-Pot itfomt, with applicatio i Lida dtctio, spctal imagig ad optical tlcommuicatio. SCPC dvics woig i both visibl ad a ifad (NIR) wavlgth ags a dmostatd. A adom optical switch ca b catd by itgatig a Faby-Pot itfomt with a stac of SCPC uits. iii

4 Liquid cystal Faby-Pot (LCFP) Poducts hav b aalyzd, fabicatd ad chaactizd fo applicatio i both spctal imagig ad optical tlcommuicatio. Both sigl-talo systm ad twi-talo systm a fabicatd. Fiss of mo tha i visibl wavlgth ag ad fiss i mo tha 3 i NIR a achivd fo th tuabl LCFP poduct. Th matials, fabicatio ad chaactizatio of lasig missio of dy dopd HPDLCs a discussd. Lasig fom difft mods of HPDLCs is studid ad both th switchig ad tuability of th lasig fuctio is dmostatd. Lasig fom two-dimsioal HPDLC basd Photoic Bad Gap (PBG) matials will also b dmostatd. Fially, lasig fom polaizatio modulatd gatig is discussd. iv

5 TABLE OF CONTENTS LIST OF FIGURES... ix LIST OF TABLES...xviii ACKNOWLEDGEMENT... xix CHAPTER I. INTRODUCTION TO LIQUID CRYSTALS... Physical Poptis of Liquid cystals... Liquid cystal phass... Aisotopic poptis of liquid cystals... 5 Th Fa F Egy ad th cotiuum thoy... 9 Sufac Aligmt of Liquid cystal... Modlig of Dicto Cofiguatio of Liquid Cystals... 3 Dicto cofiguatio i cas of ifiit sufac achoig... 3 Dicto cofiguatio i cas of fiit sufac achoig... CHAPTER. LIGHT PROPAGATION IN STRATIFIED MATERIALS... 4 Itoductio... 4 Jos matix mthod... 6 Bma's 4-by-4 matix mthod... 9 Light Popagatio i Piodic Mdia Itoductio to gatig Coupld Wav Thoy... 4 v

6 Coupld wav thoy fo tasmissio Gatigs... 4 Coupld wav thoy fo flctio Gatigs CHAPTER 3. HOLOGRAPHIC POLYMER DISPERSED LIQUID CRYSTAL... 5 Itoductio to Hologaphy... 5 Itoductio to hologaphic polym dispsd liquid cystals Tasmissio Mod HPDLCs Rflctio Mod HPDLC... 6 Vaiabl-Wavlgth HPDLC... 6 HPDLC Matials UV Mixtus Visibl Mixtus Summay CHAPTER 4. LIQUID CRYSTAL FABRY-PEROT Itoductio Itoductio to Faby-Pot itfomt Itoductio to Liquid Cystal Faby-Pot (LCFP) Tuabl Filt Fabicatio ad Tstig of LCFP Tuabl Filt Sigl LCFP systm Twi LCFP systm... 9 Eviomt Tst of LCFP Summay ad Coclusios CHAPTER 5 SWITCHABLE CIRCLE-TO-POINT CONVERTER Itoductio Bacgoud: Itoductio to HCPC vi

7 Picipl of Opatio of SCPC... Optics Dsig of SCPC... 4 Fist typ (bam stig) SCPC... 4 Scod typ (focusig) SCPC... 7 Astigmatism i scod typ (focusig) SCPC... 8 Fabicatio ad chaactizatio of SCPC woig i visibl wavlgths... 3 Sigl chal SCPC... 3 Fabicatio ad Chaactizatio of SCPC woig i NIR wavlgths... 8 Matial optimizatio fo big-aa SCPC woig i NIR... 8 Fabicatio ad Chaactizatio of sigl chal SCPC woig i NIR... Fabicatio ad Chaactizatio of 3-chal SCPC woig i NIR... 3 Summay ad Coclusios CHAPTER 6. LASING OF DYE-DOPED HPDLC Itoductio Itoductio to Dy Itoductio to las Itoductio to dy las Itoductio to Photoic Bad Gap Matials Itoductio to Lasig i Liquid Cystal Matials... 5 Itoductio to Dy-Lasig i HPDLC... 5 Dy Lasig fom HPDLC of Difft Mods: Matials, Fabicatios ad Rsults Lasig of sigl flctiv dy-dopd HPDLC Lasig of tasmissiv dy-dopd HPDLC Multipl-Mthod fo Lasig Tuig vii

8 Lasig Tuig i Stac of HPDLCs Lasig Tuig i chipd HPDLC Two Dimsioal Dy-Dopd HPDLC Lasig... 7 Lasig of Polaizatio Gatig Summay ad Coclusios CHAPTER 7 CONCLUSIONS AND CONSIDERATION ON FUTURE WORK BIBLIOGRAPHY... 9 viii

9 LIST OF FIGURES Figu Pag. Schmatic dsciptio of cystal, liquid cystal ad liquid. Schmatic dsciptio of smctic phas 3.3 Schmatic dsciptio of dicto Oitatio of Cholstic stat 3.4 Dfiitio of θ usd i quatio (-) 4.5 Od paamt S chags with th tmpatu T 5.6 Molcula stuctu of th matic liquid cystals 7.7 Dicto -oitatio i th xtal fild 8.8 Th caoical lastic distotios: (a) bd (b) Twist ad (c) Splay.9 Plaa o homogous aligmt of matic liquid cystals. Th dictio of a matic liquid cystal o a solid sufac, spcifid by th pola agl θ ad azimuthal agl φ. 3. Co-odiato systm of dicto 4. Dicto cofiguatio of 9 twist cll without applid voltag (a), ad ud applid voltag of 5V (b) 9.3 Dicto cofiguatio of a chial-dopd homotopicly aligd cll.4 Dicto cofiguatio of a chial-dopd homotopic-aligmt cll with fiit sufac achoig, with o applid voltag.5 Z-compot of th dicto of a chial-dopd homotopic-aligmt cll at difft sufac achoig stgth 3 ix

10 . Local coodiato systm E - E - z i Jos matix mthod 7. Spctal spos of cloc-wis ciculaly-polaizd light x y fom two cholstic clls 37.3 Diffactio of a moochomatic pla wav by a optically thic gatig 38.4 Diffactio of a moochomatic pla wav by a optically thi gatig 4.5 Diffactio by optically thic gatigs. (a) Tasmissio gatig; (b) Rflctio gatig 4.6 K-spac of tasmissio gatigs (a) Idal phas match Δ α = ; (b) ot idal phas match Δα Phas mismatch i tasmissio gatigs du to (a) agula dviatio; ad (b) wavlgth dviatio 45.8 Th diffactio fficicy dcass with th icas of phas mismatch Diffactio fficicy as a fuctio of phas mismatch (Δβ/) fo flctio gatigs with difft thicss L. 48. Diffactio fficicy of a flctio gatig as a fuctio of κ L fo a idally matchd phas Δβ/. 5. Modlig of flctio gatig at omal icidc. Bma s 4x4 mthod ad coupld wav thoy calculatio a compad Optical stup fo codig (a) ad costuctig (b) i-li hologam Th optical stup fo codig (a) ad costuctig (b) off-li hologam Hologaphic cofiguatios fo (a) flctiv mod ad (b) tasmissiv mod HPDLCs Tasmissio HPDLCs i th (a) o stat; (b) off stat 58 x

11 3.5 Polaizatio idpdt lcto-optical dvic basd o stacig of two polaizatio ssitiv tasmissio HPDLCs (G ad G) ad a polaizatio otato (PR) Th SEM photogaph of a tasmissio HPDLC opatig at 5 m. top substat is pald bfo SEM photogaph is ta Th SEM photogaph of a flctiv HPDLC opatig at ~5 m. Th imag is of th coss sctio of a cll Schmatic illustatio of a flctig vaiabl-wavlgth HPDLC (a) Rflctac ad pa flctd wavlgth as a fuctio of applid voltag fo vaiabl wavlgth HPDLC with a 5 μm cll gap. (b) Expimtally (poits) ad modld (cuvs) flctac spcta of vaiabl wavlgth HPDLC masud at,, ad V Diagam of a plat with factio idx immsd i th bouday mdia with factio idx of ' 69 ( t) ( i) 4. Bhavio of I / I as a fuctio of th phas diffc δ fo vaious valus of fiss Ғ Faby-Pot itfomt Imag of th Faby-Pot itfc patt with moochomatic icidt light Rlatio of th flctiv fiss with th flctivity Sphical dfcts (a), sufac igulaitis (b), ad paalllism dfcts (c) Effctiv fiss chags with th dfct fiss. FR psts th flctiv fiss Modlig of twi talo systm with th gaps of 3 mico ad mico 78 xi

12 4.9 Stuctu of liquid cystal Faby-Pot 8 4. Th avag factio idx chags with th applid voltags 8 4. Combiatio of polaizatio bam splitt ad two LCFPs with aligmt dictios ppdicula to ach oth, to achiv th polaizatio-idpdt wavlgth filtig 8 4. Two LC lays isid th Faby-Pot Cavity to achiv th polaizatio idpdt wavlgth filtig ad tuig Twist matic Faby-Pot Spctal spos of LCFP #68, masud at.5 V Spctal spos of LCFP #68, masud at 3.5 V Spctal spos of LCFP #68, masud at 9. V Elcto-optical spos of LCFP #68, masud at 85 m LCFP #68 i th housig with lctical cocto Photogaphs of th sigl talo i th housig (ight) ad th twi talo imagig filt (lft) 9 4. Tasmissio as a fuctio of wavlgth fo th 3 μm gap LCFP 9 4. Tasmissio as a fuctio of wavlgth fo th 6 μm gap LCFP Tmpatu vsus tim fo th thmal vacuum tstig of th LCFP Tasmissio of th LCFP that udwt a Pgasus-lvl sha tst fo two difft voltag sttigs ( ad 9 Volt) Tasmissio of th LCFP that udwt thmal cyclig, bfo ad aft th thmal cyclig fo two difft voltag sttigs Th ay tac diagam of th hologaphic cicula-to-poit covt (HCPC) dvlopd by McGill ad co-wos xii

13 5. Th coss-sctio dawig of a 4X switch mployig two idtical SCPC Elmts A adom optical coss-switch ca by stacig multipl SCPC uits Th fist typ of SCPC: th diffactd bam is focusd by a focal ls to a poit Radig bam cofiguatio (a) ad codig bam cofiguatio (b) of th bam stig HPDLLC fo th fist typ of SCPC Th hologaphy stup fo fabicatig th scod typ SCPC Rcodig bam pofil acoss th HPDLC aa usig th stup i Figu Th diffactio bam pofil of ich HPDLCs fabicatd usig focal lss with vaious focal lgth F 5.9 Th lft pal : th switch-off stat of th SCPC (o voltag applid); th ight pal : th switch-o stat (voltag applid). I ach pal, th hologaphic focal poit is th poit o th ight sid, ad th pass-though light is o th lft 3 5. Switchig of a SCPC woig i 53 m 5 5. A schmatic dsciptio of CAD dsig of a -pixl ITO patt i SCPC 6 5. Switchig of th ct pixl of -pixl typ-ii SCPC Switchig of o o-ct pixl of -pixl typ-i SCPC Switch o th ct pixl of a bam-stig -chal SCPC Hologaphic codig stup fo fabicatig th SCPCs woig i 55 m ag 5.6 Tasmittac ad diffactio fficicy as a fuctio of voltag xiii

14 5.7 Switchig of idpdt chals i th SCPC uit. Th photos, show that th dactivatio of th ctal pixl, th 5 th pixl (cout fom th ct), ad th outmost pixl(3 th ), spctivly Th omalizd tasmittac ad diffactio fficicy of th ct chal of a SCPC uit as th fuctio of voltag Optical stup fo masuig th wavlgth dpdc of th SCPC uits 6 5. Tasmittac ad diffactio fficicy as a fuctio of wavlgth of th switch-off stat of a SCPC sampl: JL44B 7 5. Th fittig of th modlig sult basd o coupld wav thoy ad th factio picipl, with th masumt sult, fo th wavlgth dpdc of th diffactio fficicy 9 5. Th tasmissio as a fuctio of icidt agl of th SCPC Th diffactio fficicy as a fuctio of icidt agl of th SCPC Nomalizd tasmittac is fittd to th fomula fo tasmissio gatig divd usig coupld wav thoy 3 6. Absoptio of positiv dy (a) ad gativ dy (b) Two basic ids of dys (a) azo dy (b) athaquio dy Dy molculs isid liquid cystals (a) Two-lvl gy systm of las mdium. (b) th-lvl gy systm A fou-lvl las gy diagam Molcula stuctu of th lasig dy Pyomth 58(a) ad DCM(b) Emissio spctum of a dy molcul shifts fom th absoptio spctum Littow aagmt tus of th ct pa of a las xiv

15 by otatig th diffactio gatig Lasig missio fom a flctio mod HPDLC (solid li) ad tasmissio spcta of th sam sampl (dottd li) Switchig of th dy lasig missio fom a flctio mod HPDLC Two ls w usd to gat th vtical li acoss th HPDLC gatig i od to icas th aa of th gai mdium big pumpd Lasig missio of th sampl with.5% Dy coctatio as th pump bam polaizatio is chagd fom s-polaizd to p-polaizatio Lasig missio of th sampl with % Dy coctatio as th pump bam polaizatio is chagd fom s-polaizd to p-polaizatio Lasig missio of th sampl with % Dy coctatio as th pump bam polaizatio is chagd fom s-polaizatio to p-polaizatio Dy molculs a distibutd i th liquid cystal lays ad a aligd with th liquid cystal i th sufac Lasig missio at vaious pump gis i a sampl with.5% dy Lasig missio at vaious pump gis i a sampl with % dy Pa missio itsity at vaious pump gis. A thshold at ~8 µj. Sampl has dy coctatio of.5% Effct of lctic filds o lasig i a tasmissio HPDLC. Egy of pumpig las is μj Vaious mods of opatio to tu th wavlgth pa of th lasig Stacd gatig fo tuabl lasig. Th gatig with th small pitch, low flctio bad i a zo voltag stat, whil th lag pitch gatig has a fild applid acoss it to switch off lasig. 67 xv

16 6. Tasmissio of th two gatigs usd i th stac. A is dopd with dy P58, ad gatig B is dopd with dy DCM Tuig of a chipd HPDLC. Tasmissio at lft (solid), middl (dashd) ad ight (dottd) poits (top); ad lasig missio at lft (solid), middl (dashd) ad ight (dottd) poits o th sampl (bottom) Switchig of a flctio mod chipd HPDLC (a) Stup fo catig 4-bam itfc patt ad (b) th itfc patt; th bight (da) gios pst aas of high (low) itsity SEM imag of a HPDLC lattic gatd by 4-bam itfc. Th dsigd piod is m (a) Stup fo catig 6-bam itfc patt (b) th itfc patt; th bight (da) gios pst aas of high (low) itsity (a) Isoitsity plot fo fou-bam fabicatio ; ad (b) lasig fom this stuctu dopd with th las dys Pyomth 58 (solid li) ad DCM (dottd li). Lasig missio is masud alog x-dictio (a) Isoitsity plot fo six-bam fabicatio ad subsqutly lasig; ad (b) lasig fom this stuctu dopd with th las dys Pyomth 58 (solid li) ad DCM (dottd li) a) Two lialy polaizd bams with othogoal polaizatio dictios; (b)two ciculaly polaizd bams with opposit ss of clocwis Micoscop imags of a cll of polaizatio gatig btw polaizs. (a) o voltag is applid; (b) V voltag is applid Witig bam ad pump bam fo fabicatio ad lasig missio tstig of th polaizatio gatigs 83 xvi

17 6.33 Lasig missio fom a liquid cystal polaizatio hologaphy gatig Thshold of las missio fo th dy-dopd polaizatio gatig lasig missio icass by 5% as th icidt polaizatio is otatd Effct of a applid lctic fild o a liquid cystal polaizatio gatig pumpd by p-polaizd light. 87 xvii

18 LIST OF TABLES 4. Fiss ad f spctal ag of LCFP # 68 at difft voltags Tstig sult of tuabl LCFP fo tuabl las i NIR ag Covgig codig bam icidt agl, Bagg adig ad diffactio agls, diffactio agls with omal icidt adig, ad miimum distac btw ighboig SCPC uits. 5. Tstig sult of SCPC Compots of th HPDLC mixtus iitially ivstigat 5.4 Matial cotts of Fomula-SCPC 5.5 Th tasmittac of som chals of a SCPC uit 5 xviii

19 ACKNOWLEDGEMENT I would li to tha my two advisos, Pofsso Ggoy P. Cawfod ad Pofsso Jac Klly, fo all thi ducatio, istuctio, ad suppot. Duig th th yas I studid ad wod i th Liquid cystal Istitut of Kt Stat Uivsity fom 996 to 999, I had lad a lot fom all th pofssos ad tachs. I would li to xpss my gatitud to all of thm. I would li to tha Scitific Solutios, Ic., th compay I hav wod fo sic lat, fo povidig a gat sach platfom fo m to cotiu my PhD sach. I also tha th Display ad Photoics lab of Bow Uivsity fo th happy coopatio o all th sach pojcts I hav do duig ths yas, spcially, with spcial gatitut to Pofsso Ggoy P. Cawfod who has giv m so may suppot, dictio ad ispiatio. I got a lot of hlp fom D. Haiqig Xiayu, M. Scott Woltma (PhD cadidat), M. Jiahua Lia, D. Ju qi, ad D. Matthw Sousa fom Display ad Photoics lab of Bow Uivsity. I would li to tha all of thm fo thi assistac ad hlpful discussios. Spcial gatitud to my family, spcially my pats, fo thi log-tm suppot ad couagmt. I would li to us my disstatio as a spcial gift to my lovly daught, hopfully sh will li it mo tha a toy. Fially, I would li to tha my Small Busiss Iovatio Rsach (SBIR) pojct sposos: Natioal Scic Foudatio, NASA ad Dpatmt of Egy. xix

20 CHAPTER I Itoductio to Liquid Cystals. Physical Poptis of Liquid Cystals Th liquid cystal phas is a itmdiat stat btw th solid cystalli phas ad th isotopic liquid phas (Fig..). Th distiguishig chaactistic of th liquid cystalli stat is th tdcy of th molculs (msogs) to poit alog a commo axis, calld th dicto. This is i cotast to molculs i th liquid phas, which hav o itisic od. I th solid stat, molculs a highly odd ad hav littl taslatioal fdom. Th chaactistic oitatioal od of th liquid cystal stat is btw th taditioal solid ad liquid phass; this is th oigi of th tm msogic phas, usd syoymously with th liquid cystal stat. Liquid cystals xhibit som dg of fluidity, which may b compaabl to that of a odiay liquid; Howv thy also xhibit aisotopis i thi optical, lctical, magtic ad oth physical poptis li cystals... Liquid Cystal Phass Liquid cystal phass a obsvd i ctai ogaic compouds ad usually a mad up of logatd molculs. Th a a umb of distict liquid cystal btw th cystalli phas ad th isotopic liquid. Ths itmdiat tasitios may b bought about by tmpatu vaiatio; th compouds i which th liquid cystal phas is iducd by a thmal pocss a ow as thmotopic liquid cystals. Th thmotopic liquid cystals a futh classifid ito th typs: matic, smctic ad cholstic as poposd by Fidl []. This classificatio is

21 basd o th molcula aagmt ad th odig of th molculs i th paticula liquid cystal phass. Nmatic liquid cystals hav log-ag oitatioal od but o log-ag taslatioal od. Th avag oitatio of all of th molculs i th matic liquid cystals is dfid as th dicto, as show as th aow i Fig..(b). Smctic liquid cystals (Fig..) a difft fom matics i that thy hav a additioal dg of positioal od. Smctics gally fom lays withi which th is a loss of positioal od, whil th oitatioal od is still psvd. Th a sval difft catgois to dscib smctics. Th two bst ow of ths a Smctic A, i which th molculs td to alig ppdicula to th lay plas, ad Smctic C, wh th aligmt of th molculs is at som abitay agl to th omal. (a) (b) (c) Fig.. Schmatic dsciptio of cystal (a), liquid cystal (b) ad liquid (c). Th cholstic phas (o chial matic phas) is typically composd of matic msogic molculs cotaiig chial ct that poducs itmolcula focs, which favo a aligmt btw molculs at a slight agl to o aoth. This lads to th fomatio of a stuctu that ca b visualizd as a stac of vy thi -D matic-li lays with th dicto i ach lay twistd with spct to thos abov ad blow (Fig..3). I this stuctu, th dictos actually

22 3 fom a cotiuous hlical patt about th lay omal. Th blac aows i th Fig..3 pst th dicto oitatio i th succssio of lays alog th stac. A impotat chaactistic, th pitch, p, is dfid as th distac it tas fo th dicto to otat o full tu. (a) (b) Fig... Schmatic dsciptio of smctic phas. smctic A phas (a); smctic C phas (b). Fig.3. Schmatic dsciptio of dicto oitatio of cholstic stat. Aows pst th dicto dictios i ach lay.

23 4 ˆ θ Fig..4. Dfiitio of θ usd i quatio (-). Nmatic liquid cystals a usually uiaxial ad a th most widly usd liquid cystals i lcto-optical applicatios such as fo th twistd matic ffct, phas modulatio, tc. Th dicto dtmis th dictio of th pfd oitatio of th molculs but dos ot pst th dg of th oitatioal od. Th od paamt S, poposd by Tsvtov [], povids us with a masu of th log ag oitatioal od: 3cos θ S = (-) wh θ is th agl btw th axis of a molcul ad th dicto of th liquid cystal (Fig..4); th agula bacts idicat a avag ov th complt systm. Fo a pfct cystal S = ad fo th isotopic phas S =. Fo matics, S will hav a valu btw ad, vayig with th tmpatu. Th citical tmpatu at th matic to isotopic tasitio poit is dfid as T (Fig..5).

24 5.. Aisotopic Poptis of Liquid Cystals Liquid cystals show aisotopy i thi magtic, lctical, optical ad oth physical poptis. A macoscopic aisotopy is foud i liquid cystals bcaus th molcula aisotopy dos ot avag to zo, as is th cas i a isotopic phas. Fo th itst of this thsis w oly discuss thi lctical ad optical aisotopis. I uiaxial matic liquid cystals th dilctic tso ε ca b diagoalizd with igvalus ε // ad ε, which f to th dilctic costats paalll ad ppdicula to th matic dicto ˆ, spctivly. Th dilctic aisotopy is witt as Δε = ε // ε (-) ad th dilctic tso is dfid as: ε αβ = ε δ + Δε (-3) αβ α β wh ad a th compots of th dicto ˆ. α β. S.5 T T Fig..5. Th od paamt S chags with th tmpatu T.

25 6 Th thotical cosidatio [3] suggsts that ε // ρ α // = ε + 3ε (-4) ε ρ α = ε + 3ε (-5) wh ε // + ε ε = 3 is th avag dilctic co-fficit that dos ot dpd o th od paamt. α // ad α a th avag molcula polaizability wh th applid fild is paalll o ppdicula to th dicto, spctivly. Fo molculs without pmat dipols, α // = α + ΔαS (-6) 3 α = α ΔαS 3 (-7) wh S is th od paamt ad α = α + α ) / 3 (-8) ( // Δα = α // α (-9) α // ad α a th molcula polaizability alog th log molcula axis dictio o ppdicula to th log molcula axis, spctivly. Fo molculs with pmat dipols, (-6) ad (-7) should b modifid to iclud th dipol tm: μ p // p p // + μ μ μ p α ( // = α + + Δα + ) S (-) 3KT 3 KT

26 7 wh μ p // o μ p μ p // p p // + μ μ μ p α ( = α + Δα + ) S (-) 3KT 3 KT a th compots of pmat dipol μ alog th log-molcula axis o ppdicula to th log-molcula axis, spctivly. C7H5 N N C7H5 (a) C7H5 CN (b) OC6H5 CH5O C N (c) Fig..6. Molcula stuctu of th matic liquid cystals: (a) a o-pola liquid cystal molcul; (b) a pola liquid cystal molcul with positiv dilctic aisotopy; (c) a pola liquid cystal molcul with gativ dilctic aisotopy.

27 8 Fom (-) ad (-) w ca s that wh th is a lag agl btw th pmat dipol ad th log molcula axis dictio, μ p // < μ p ad gativ; thfo, α // < α ad fom (-4) ad (-5) w fid ε // μ p // μ p Δα + may b KT ε o Δε <. < Fig..6 shows th ids of matics liquid cystals. (a) is a o-pola molcul whil (b) is a pola molcul with a dipola momt paalll to th log molcula axis, thus Δε ( b) > Δε ( a) >. I molcul (c) th CN goup itoducs a lag pmat dipol momt at a lag agl with th log molcula axis dictio, so Δε ( c) <. Th dilctic aisotopy itoducs body toqu o th molculs i th psc of a xtal fild, which i tu givs is to th dicto -oitatio. (Fig..7) This popty ca b usd fo liquid cystal matials with both positiv ad gativ dilctic aisotopis. Ud a xtal fild, th dicto of a liquid cystal with a positiv dilctic aisotopy tds to alig paalll to th xtal fild, whil th dicto of a liquid cystal with a gativ dilctic aisotopy tds to alig ppdicula to th xtal fild. Elctic Fild Fig..7 Dicto -oitatio i th xtal fild

28 9 Liquid cystals a also foud to hav optical aisotopy, o bifigc, du to thi aisotopic atu. Thy dmostat doubl factio, o light polaizd paalll to th dicto has a difft idx of factio (that is to say it tavls at a difft vlocity) tha light polaizd ppdicula to th dicto. Th optical aisotopy, o bifigc is giv by: Δ = o (-) Wh is th odiay idx of factio, ad is th xtaodiay idx of factio. Th o latio btw th optical aisotopy ad th dilctic aisotopy is giv by: ε = o ; // ad ε = (-3). Th Fa F Egy ad th Cotiuum Thoy I a liquid cystal systm, th bul f gy of a ihomogous sampl has cotibutios fom th lastic dfomatio of th systm. Th lastic poptis of liquid cystals ifluc th bhavios of ths matials i a lctic o magtic fild. Th simplst way to tat th dfomatio of a matic liquid cystal is to cosid it to b a cotiuous lastic mdium, disgadig th dtails of th molcula stuctu. Th stat of th systm is dscibd by th dicto fild (), which dtmis th lastic f gy of th systm. Th stiffss of th systm ca b xpssd by a fouth a tso [4]: F l 3 = d xkijl i j l, wh K ijl is a tso that gally dpds o th local dicto (). Cosidig th symmty of th matic liquid cystal, th f gy should b ivaiat ud th symmty opatio. is a uit vcto; thfo, i ji is zo. Ths factos idicat that wh th bul gy is cosidd, K ijl has th idpdt compots, which ca b dsigatd as lastic

29 costats,,, as dpictd i Fig..8. Th fist distotio, splay, is dscibd by 33. Th scod id of distotio, twist, is dscibd by ( ). Th thid distotio, bd, is valuatd by ( ). I th cotiuum thoy, fist statd by Os [5] ad Zoch [6], ad compltd by Fa [7], th Fa f gy dsity of a matic liquid cystal mdium with a cuvatu dfomatio i its dicto fild is f = { ( ˆ) ( ˆ ( ˆ)) ( ˆ ( ˆ)) } (-4) Wh, ad cospod to th lastic costats of splay, twist ad bd, spctivly. 33 Th sufac lastic costats hav b igod i (-4); thy td to play a lag ol i highly cofid liquid cystal systms [3]. This fom of th Fa f gy dsity is miimizd wh th dicto is spatially uifom. (a) (b) (c) Fig..8. Th caoical lastic distotios: (a) bd (b) twist ad (c) splay.

30 Fo cholstic liquid cystals, th a spotaous twists, which a oigiatd by th chial molculs. A additioal tm that tas ito accout th chiality of th molculs is itoducd i th scod tm of (-4) sultig i th xpssio: f = { ( ˆ) ( ˆ ( ˆ) ) ( ˆ + + q + 33 ( ˆ)) } (-5) wh q = π / p is th wav vcto; p is th pitch of cholstic. Positiv ad gativ valus of q cospod to a lft o ight-hadd hlix, spctivly..3 Sufac Aligmt of Liquid Cystal I may liquid cystal dvics, such as twistd matic clls ad wavplats, a uifom o wll-dfid oitatio of th liquid cystal molculs is quid. Without sufac aligmt ad cll cofimt, th liquid cystal cll will hav multipl domais with difft oitatios, ad bouday walls ad dfcts btw th domais. Th multi-domai atu ad th xistc of umous bouday walls ad dfcts sult i stog scattig. Spcially tatd sufacs a mployd i od to su a sigl domai i th dsigatd aa. (a) ( b) Fig..9. Plaa o homogous aligmt (a) ad homotopic aligmt (b) of matic liquid cystals.

31 Two typs of sufac aligmt, as show i Fig..9, a widly usd i liquid cystal dvics, distiguishd by th pfd oitatio of th molculs o th sufac. With plaa o homogous aligmt, th molculs a oitd i a dictio paalll to th sufac; whas with homotopic aligmt, th molculs a oitd i a dictio ppdicula to th sufac. Plaa aligmt ca b achivd by uidictioally ubbig a coatd polyimid lay [8], o by xposig photo-aligabl polyimid to polaizd UV light [9]. Homotopic aligmt is alizd by dpositig amphiphilic molculs such as lcithi [], sila [], o som polyimids, such as SE-75 fom Bw Scic, o th sufac. A sufac achoig tm is itoducd ito th f gy with th cosidatio of th aligmt ffct. I th viciity of th tatd sufac, th is a gtically favoabl dictio giv by a uit vcto. I th modl pstd by Rapii ad Papoula, th sufac f gy dsity is giv by []: f suf = w( s ) = wsi ( θ ) + cost (-6) wh w is th achoig stgth, is th dicto at th sufac; ad θ is th agl s 4 7 btw ad. Th typical valu of w is i th od of ~ J/ m [3]. s Wh a lctic fild is applid to a matic liquid cystal cll, th spatial molcula cofiguatio ca b dtmid by miimizig th f gy of th systm: F = F + F + F l fild suf = 3 d x{ K ( ) + K [ ( )] + K [ ( )] } 3 ds[ wsi ( θ )] 3 + d x[ Δε ( E ) ] + (-7)

32 3 θ s Fig... Th dictio of a matic o a solid sufac, spcifid by th pola agl θ ad azimuthal agl φ..4. Modlig of Dicto Cofiguatio of Liquid Cystals Th basic cocpt of dicto cofiguatio modlig is to fid th dicto cofiguatio that miimizs th total f gy of th systm. Th total f gy icluds th bul tm, which is dscibd as th Fa-Os stai f gy, th sufac tm, which is sufac f gy, ad th tm latd to th xtal lctic fild. Fo th sufac tm, both th two cass a discussd: ifiit sufac achoig ad fiit sufac achoig..4. Dicto Cofiguatio i Cas of Ifiit Sufac Achoig Wh th sufac achoig gy is stog ough to b tatd as ifiit, th f gy dsity quatio fo a liquid cystal matial i a lctic fild, basd o th Fa-Os stai f gy dsity, is giv by [7]: f ) ) ) ) ) ( ) + [ ( ) q ] + [ ( )] 33 = { } ± D E (-8)

33 4 wh ˆ is th dicto ad q = / p, p is th atual pitch of th matial; ad,, π 33 a th splay, twist, ad bd lastic costats, spctivly. I th ± D E tm, D is th lctic flux dsity ad E is th lctic fild; + is fo th cas of costat lctic flux dsity ad is fo th cas of costat lctic fild. I ou applicatio w usually cosid th applid voltag as a costat, so w coctat o th latt costat voltag coditio. Assumig th dicto oly chags alog th cll omal, dfid as th z-axis, a odimsioal coditio, w us th coodiat systm show i Fig... z z x y x y Fig... Co-odiato systm of dicto It is asoabl to assum th is o f chag i th liquid cystal; fom Maxwll s Equatio dd D = w hav z =. Also, cosidig th costat voltag, dz Edz = V (-9)

34 5 So fom Dz dz = V ε [ ε // z + ε ( z )] (-) W obtai D z = d ε // z ε V dz + ε ( z ) (-) This mas th omal compot of D is costat thoughout th cll. Th f gy ca ow b witt as: f ( ) [ ] [ ( ) ( ) & z + x& y + y& x q + 33 z & x + & y + x& x + y& y = { ]} + ε [ ε // z D + z ε ( z )] (-) with us & pstig d dz. Th total f gy is giv by itgatig (-) ov th volum: F = A d t fdz (-3) Th f gy fuctio is F = d Dz d A f D x + y + z dz A f dz { λ λ ( )} (-4) ε [ ε + ε ( )] // z z wh λ D is th LaGag multipli fo th costait of costat voltag, ad λ is aoth LaGag multipli fo th costait ˆ =. Now th poblm bcoms th d to fid th dicto cofiguatio ˆ = ˆ( z) which will miimiz th total f gy fuctio i (-4) ad mts th bouday coditios. Th statioay coditio lads to th Eul-Lagag quatios. Fistly

35 6 )] ( [ // = + = z z D z z D f D f ε ε ε λ δ δ δ δ, sultig i: D D z = λ (-5) Cosidig (-) ad (-5) w obtai: ]} ) ( ) ( [ ] [ ) ( { 33 y y x x y x z x y y x z q f & & & & & & & = )] ( [ // z z z D + ε ε ε ) ( z y x + + λ (-6) Aoth Eul-Lagag quatio is: ) ( = i i i f dz d f f & δ δ, i = x, y, z (-7) W d to solv (-7) to fid th dicto cofiguatio of th quilibium stat. If w choos a sphical co-odiat systm i which th paamts θ ad φ a usd, th costait ˆ = ca b automatically satisfid, but wh th dicto is alog th z dictio, φ ca b ay valu, ad this lads to cofusio. I ou modlig w istad us th paamts,,. x y z Istad of solvig (-7), w us th laxatio mthod basd o th dyamic quatios of th dicto to fid th dicto cofiguatio of th quilibium stat: )] ( [ i i i i f dz d f f t & = = δ δ γ, i = x, y, z (-8) wh γ is a viscosity cofficit. Disctizig ths quatios givs: i i f t δ δ γ Δ = Δ (-9)

36 7 I dtail, w hav: Δ = Δ ] ) ( ) [ ( { y x y y x y x y y x x q t & & && & & & γ (-3) } ] ) ( [ 33 x x y y y x x x x z z x z λ & && & && & & && Δ = Δ ] ) ( ) [( { x x y y x x x y y x q t y & & && & & & γ (-3) } ] ) ( [ 33 y y y y y x x x y z z y z λ & && & && & & && } ] ) [( ) ( ) ( { // // 33 z z z z y x z z z D t λ ε ε ε ε ε ε γ Δ = Δ & & & & (-3) At ach tim stp, is updatd by i i i Δ +, ad is also upatd as i (-). D z W ca glct th λ tm i f xpssio (-6), if w -omaliz at ach tim stp of th laxatio. Th, th dicto at th tim stp + is: + i ) ( i i i f t δ δ γ Δ = + (-33) + + = + i i (-34) Th itatio pats util th dicto covgs to th quilibium stat. Fig.. is th calculatd dicto cofiguatio of a 9 twist liquid cystal cll (with plaa bouday coditios). Fig..(a) is ud o voltag ad Fig..(b) is ud 5V voltag. Th followig paamts usd a: thicss d=5. μm, ptilt agl p θ =, lastic costat: =5.5(pN), =4., =8., 33 // ε =8., ε =3.3

37 8 Fig.3 is a calculatd dicto cofiguatio of a chial-dopd liquid cystal cll with homotopic bouday coditios. Fig..3 (a) is ud v voltag ad Fig..3 (b) is ud V voltag. Th paamts usd a: thicss d=5. μm, ptilt agl θ p = 9 (homotopic bouday), lastic costat: = 4.9 (PN), =7.9, =5., ε =3.3, ε =8., th d/p atio is. 33 //

38 9..9 x Compot of Dicto y. z Nomalizd Thicss (a). z.9.8 Dicto Compot x y Nomalizd Thicss (b) Fig. (a) Dicto cofiguatio of 9 twist cll without applid voltag. (b) Dicto cofiguatio of 9º twist cll ud applid voltag of 5V.

39 ..8 z Dicto Compot y x Nomalizd Thicss (a)..8.6 y Dicto Compot x z Nomalizd thicss (b) Fig..3 Dicto cofiguatio of a chial-dopd homotopically aligd cll. (a) o voltag applid; (b) V applid voltag.

40 .4. Dicto Cofiguatio i Cas of Fiit Sufac Achoig Wh th achoig stgth of th sufac aligmt is ot ifiitly stog, bsids cosidig th f gy i th bul, which is dscibd by (-7), w also d to cosid th sufac achoig gy. Accodig to th dsciptio of Rapii ad Papoula [], th sufac achoig gy is: f a ) ) ( ˆ) = Wa[ ( ) ] (-35) Wh Wa is th sufac achoig stgth; ) is th asy dictio fo th sufac dicto; ad ) is th actual sufac dicto which dviats fom ). Th sufac achoig gy is miimizd wh th dicto is aligd alog th asy dictio. Th total f gy is: F = f dv ) Wa ( ) ds f ) dv + gds (-36) Not that th itgatio of th scod tm is ov th sufac. Accodig to th Eul-Lagag quatio, th quilibium stat satisfis: δf δ i f i d dz f ( ) = & i ( fo th bul ) (-37) g f Ad = & i i (fo th sufac z=) (-38) g f + & i i = (fo th sufac z=d) (-39) Wh i is th idx fo x,y,z. Th latt two quatios a th so-calld toqu balac quatios. Cosidig th dyamics i th laxatio mthod, w obtai: i δf f d f γ = ( ) (fo th bul) (-4) t δ dz & i i i

41 Ad g f i γ s = (fo th sufac z=) (-4) t i & i g f i γ s = + (fo th sufac z=d) (-4) t i & i Wh γ s is th viscosity costat of th sufac.. y Dicto Compot z x Nomalizd Thicss Fig..4. Dicto cofiguatio of a chial-dopd homotopic-aligmt cll with fiit sufac achoig (o applid voltag). Fig..4 shows th calculatd dicto cofiguatios of a chial-dopd liquid cystal cll with homotopic sufac aligmt. Th achoig stgth is 8. 6 J / m. Th oth paamts a: d/p =.6, thicss d=5. μm, = 4.9 (pn), =7.9, =5.. Bcaus th sufac achoig stgth is fiit, th bul twistig stgth, which tds to tilt th dicto away fom th cll omal, is lativly stog; th 33 z is dcasd with th icas of sufac

42 3 achoig gy. Fig..5 plots th z compot of th dicto though clls with difft sufac achoig stgths. Th oth paamts a th sam as thos paamts usd i Fig z compot of dicto--z Blac: Wa = 5.-6 Blu: Wa = 7.-6 Rd: Wa = 8.-6 G: Wa = Nomalizd Thicss Fig..5. Z-compot of th dicto of chial-dopd homotopic-aligmt clls at difft sufac achoig stgths.

43 CHAPTER Light Popagatio i Statifid Matials. Itoductio I classical lctomagtic thoy, th stat of light is dscibd by two vctos, th lctic fild E ad th magtic iductio B. Light popagatio is dscibd by Maxwll's quatios [4]. B E + = t D H = j t D = ρ B = (-) wh D th lctic flux dsity, H th magtic fild stgth, j th cut dsity, ad ρ is th chag dsity. Th poptis of th mdium though which th light is popagatig a also cssay to dtmi th fild distibutio: j = σe, D = εe, H = B, μ (-) wh σ is th coductivity, ε th dilctic costat, ad μ is th magtic pmability. σ, ε ad μ a tsos flctig th poptis of th matials ad may dpd o E ad B. I a dilctic mdium, σ is gligibl i most situatios. 4

44 5 Wh discussig light popagatio i dilctic mdia, w usually assum th is o souc i th mdia, i.. o f chag ( ρ = ) ad o f cut ( j = ). I a homogous ad isotopic dilctic mdium, th followig quatios a dducd fom Maxwll's quatios: E E εμ t H H εμ t = =,. (.3) Ths a th stadad wav quatios with moochomatic pla wav solutios: wh is th wav vcto ad is: E = H = E H i( ωt ), i( ωt ), (.4) ω is th agula fqucy. Th latioship btw ad ω = ω =, c (.5) wh εμ is th factiv idx. Both th lctic fild E ad th magtic fild H a ppdicula to th popagatio dictio ( E = ), ad a ppdicula to ach oth. Th phas vlocity of light is dscibd by: = H v = c c =. εμ (.6)

45 6 Th dilctic costat ε = ε( ω) dpds o th fqucy of th lcto-magtic wav. Thfo, lcto-magtic wavs of a ctai fqucy popagat i a mdium at thi ow phas vlocity, a phomo fd to as dispsio i optics. Wh cosidig light popagatio i liquid cystal dvics, i most cass ths matials ca b tatd as statifid aisotopic matials. Th basic ida is to divid a liquid cystal dvic ito may lays; if lay umb is lag ough, ach lay is assumd to b optically uifom. Two widly accptd mthods, th Jos matix mthod ad Bma's 4x4 mthod [5] a itoducd h.. Jos Matix Mthod I th Jos matix mthod, th polaizatio stat of light is dscibd by a Jos vcto E E = E x y (-7) wh E, E a th complx compots of th lctic fild i th x, y dictios, x y spctivly. Th light is assumd to popagat alog th z dictio. As th amplitud ad th phas diffc btw th two compots a cosdd, so th vcto is -witt as: Ex Ex E = = iφ (-8) E y E y wh φ = π( o )d/λ is th phas diffc of th o-light ad th -light. Th light itsity is giv by: I = E x + E y (-9) Wh light passs though a uifom bifigt film, a Jos matix, J, is usd to dscib th optics of th systm:

46 7 wh E, E i out E J = (-) out E i a th icidt light ad th tasmittd light, spctivly. Fo a o-absobig bifigt film with a phas shift ϕ ad i which th dictio of th fast axis (alog which dictio th factio idx is th smallst) mas a agl θ with spct to x-axis, th Jos matix is : J = R( θ ) J ϕ R( θ ) = R( θ ) J ϕ R( θ ) (-) cos( θ ) si( θ ) Wh Jϕ = ad R( θ ) =. W may tat a matic liquid cystal iϕ si( θ ) cos( θ ) cll as may bifigt lays, so th total dvic ca b pstd by o Jos matix J total J total = J J (-) m m... J J z z x y E y x E x y Fig.. Local coodiat systm E - E - z i th Jos matix mthod. x y

47 8 Th Jos vcto of th output light ca th b calculatd: E out E E = J E E out i x x = out total i y y (-3) I umical computatio, wh th liquid cystal has a twist stuctu, it is covit to assum a local co-odiat systm, i which th E x axis is hld to b th sam as th i-pla compot of th dicto, ad th E y axis is hld i th x-y pla (Fig..). It is impotat to mtio that th E axis is ppdicula to th E -z pla, so E is othogoal to th dicto. y x y Suppos th dicto i th lay is ) ˆ = (,, ( ( ) ( ) ( ) x y z ), th lctic fild i th local lctic fild co-odiat systm of th lay isth: ( ) E ( ) x E out = ( ) (-4) E y Wh light passs though th + lay i which th dicto is ˆ is: = (,, ( + ) ( + ) ( + ) ( + ) x y z ), th lctic fild i th local co-odiat systm of th + lay E ( + ) out E E ( + ) x = ( + ) y i = ϕ cos( θ ) si( θ ) si( θ ) E cos( θ ) E ( ) x ( ) y (-5) wh ϕ is th phas diffc of th -wav ad th o-wav: π ϕ = ( d (-6) λ ff o ) ad ff is th ffctiv factiv idx of th -wav i th + lay: o o ( + ) z ff = (-7) [ ( ) ] + ( ) ( + ) z

48 9 Th Jos matix mthod is a -by- matix mthod; it is asy to b pogammd ad applid. Wh cosidig th light popagatig fom o lay to aoth, flctio is ot cosidd; omally th Jos matix mthod is ot usd to calculat flctac..3 Bma's 4-by-4 Matix Mthod Bma s 4-by-4 matix mthod was iitially applid to liquid cystals i 97 [5]. It is basd o th solutio of Maxll s quatios without ay sigificat appoximatios; it is a accuat way to dscib th optical poptis fo ay mdium as log as th mdium ca b tatd as a multi-lay stuctu ad i ach lay th optic axis is uifom. A isotopic mdia lay is also tatd th sam way. Bma s 4-by-4 mthod has b xtsivly usd i th liquid cystal display (LCD) modlig, fo th calculatio of tasmissio, flctio, spctum ad chomaticity, cotast-viwig agl ad oth optical poptis. It is usd i most commcializd LCD modlig softwa. I this mthod, a Catsia-coodiat systm is usd i which th liquid cystal cll is i th x-y pla. Th dilctic tso is assumd to b oly a fuctio of z. Cosidig a obliquly icidt light o a sampl with a wav-vcto = (,, ), th lctic fild x y z E = i( ωt ) E ca b witt as : E = E i z i( ωt xx y y) Z = E ( z) i( ωt xx y y) (-8) Similaly th magtic fild is dscibd by: H = H i z i ( ω t x x y y ) Z = H ( z ) i ( ω t x x y y ) (-9) I od to ma th filds dimsiolss, w chag vaiabls such that: =, = E H =, h E H ε ij, ε ij = ε

49 3 Aft cosidig Maxwll s quatios: E i H ωε = (-) H i E = ωμ (-) W obtai: iax z X = (-) wh Ad = y x y x h h X A is th difftial popagatio matix = zz x xz zz y xz zz zy zx xy x y zz zx xx y zz x yz zz y yz zz zy yy x zz zx yz yx y x zz y x zz y zz y zy zz y zx zz x zz y x zz x zy zz x zx A ε ε ε ε ε ε ε ε ε ε ε ε ε ε ε ε ε ε ε ε ε ε ε ε ε ε ε ε ε ε ε ε ε ε (-3) To simplify th quatio, A is diagoalizd to : S A S A = ; with 4 igvalus λ, λ, 3 λ, 4 λ, which pst th fowad ad bacwad popagatig odiay ad xta-odiay wavs xplicitly witt as: (-4) / ) ( m = ε λ ( / / o f t t y x t x m m ε ε ε ε ε λ + + = ) (-5)

50 3 λ 3 = - λ (-6) ( m / λ = ε m ε ε ε ε ) (-7) x t / 4 t t f o x + y Th fou associatd igvcto will b giv as: ψ,3 = si( a cos( z y x mz / λ,3 )) m z λ,3 x ε o y / λ,3 (-8) m mzλ,4 ( ) x ε o ε o ψ =,4 y (-9) si( a cos( z )) λ,4 y λ,4x mz wh = (-3) ε t z z ε o + ε f y z x ε o + ε t ε = (-3) m t ε t = m( ) ctg( a cos( z )) (-3) ε o x m = = si( θ i ) (-33) ad whθ i is th agl of icidc i vacuum. A spcial cas should to b mtiod is wh z =; th th fou igvalus ad th fou igvctos caot b usd. Thy a istad dfid as:

51 3 (-34) / ) ( m = ε λ ) ( o m ε ε λ = (-35) 3 λ = - λ (-36) 4 λ = - λ (-37) (-38) =,3,3 λ ψ (-39) = /,4,4 ε o λ ψ Th matics S ad a giv by: S ),,, ( 4 3 ψ ψ ψ ψ = S (-4) (-4) M S N S T = wh is th taspos of S ad T S (-4) = M (-43) = ψ ψ ψ ψ ψ ψ ψ ψ M M M M N T T T T

52 33 Now, (-43) bcoms: Y = ia Y (-44) z Wh Y = S X. This quatio ca b solvd aalytically by: Y ( z + Δ i z z) = Δ Λ Y ( z) (-45) i So ( ) ( ΔzΛ X z + Δz = S S ) X ( z ) (-46) iδzλ iδzλ iδzλ wh = (-47) iδzλ3 i Δ zλ4 I th umical computatio, w divid th sampl ito slabs, ach of which has a thicss Δz, ad Δz is small ough fo th dilctic tso i ach slab to b tatd as costat. Th filds that xit th sufac a giv by: X = b (-48) out X i b i ΔzΛ iδzλ = ( S S ) *...*( S S ) (-49) W slct a coodiat systm i which th wav vcto is i th x-z pla, so y =. ) Cosidig a liquid cystal matial whos dicto is =,, ), th th dilctic tso is giv by: ( x y z ij o ij o ε = δ + ( ) i j (-5) H, a th odiay ad xtodiay factio idics if th matial is a pu liquid cystal. o

53 34 I od to mt th bouday coditios, cosidig th flctio of light, th fild at th iput sufac is: wh X i is th icidt fild ad X X i = X + X (-5) i is th flctd fild at th iput sufac. Th tasmitd light is dscibd by : X t = b X + bx (-5) i Th wav vcto of th icidt, flctd ad tasmittd wavs a latd as: wh N ) is th sufac omal ad ) ) = ( N) N (-53) i i i t = if th sampl is suoudd by th sam isotopic mdium at th iput ad th output sufacs. Th magtic fild compots ca b witt i tms of th lctic fild, sic B E = = iωμh t (-54) W hav E = ωμh (-55) O ) ωμ = h = zh Z (-56) wh Similaly E Z =, H E Z = ad H z = Z Z ) v h = (-57) z Fom (-56) ad (-57), w ca solv fo h ad h i tms of ad. I gal, x y x y

54 35 = y x z y x z y z x z y x y x y x z z z z h h (-58) Giv a icidt light fild, th tasmittd ad flctd light filds a: = iy ix iy ix iy ix h h α, = ty tx ty tx ty tx h h β ad = y x y x y x h h γ (-59) wh = iz i iy ix iz i iy iz i ix iz i iy ix z z z z α (-6) = tz t ty tx tz t ty tz t tx tz t ty tx z z z z β (-6) = z y x z y z x z y x z z z z γ (-6) Fom (-58) w gt:

55 36 + = y x iy ix ty tx b b γ α β (-63) Notig = y x ty tx ty tx β β ad = y x ty tx y x γ γ, W hav = ) ( iy ix y x ty tx b b α γ β (-64) So, = ) ( iy ix y x ty tx b b α γ β (-65) Now, fom (-65), w ca calculat both th tasmittd ad th flctd lctic fild, ad th magtic fild will also b calculatd. Th light itsitis of both th tasmittd ad flctd light will b ow. Fig.. shows th tasmittacs of ciculaly-polaizd light fom two cholstic liquid cystal clls. Fo th cll whos twist ss is idtical to th haddss of th ciculalypolaizd light, th is a flctio bad with a bad width of ( - )p ad a flctio wavlgth ct at ( + )p/. Th paamts usd h a: thicss d = 6. μm, ptilt agl o o p θ =, d/p atio = 5, lastic costat: = 5.5 (PN), =4., =8., =.5, =.7. Th tac ad xit mdia a ai (=). 33 o

56 Tasmittac ( % ) Wavlgth (m ) Fig... Spctal spos of cloc-wis ciculaly-polaizd light fom two cholstic clls. Blac: twist ss of th cll is clocwis; Rd: twist ss of th cll is cout-clocwis.4 Light Popagatio i Piodic Mdia.4. Itoductio to Gatig Gatigs a piodic mdia that hav b widly usd i optical applicatios. A gatig ca fall ito o of two catgois: a amplitud gatig, which spatially modulats th itsity of light, ad a phas gatig, which spatially modulat th phas of light. Th latt ca b alizd by piodically modulatig th thicss o th factiv idx of a mdium, as i hologaphic polym dispsd liquid cystals (HPDLC), which a th focus of ou sach. I a piodic mdium, th factiv idx xhibits a taslatioal symmty:

57 38 ( ) = ( + a), (-66) wh a is a costat vcto. Fo a o dimsioal phas gatig, th factiv idx satisfis ( z) = ( z + mλ), wh m is a itg ad Λ is th piod of th gatig, o pitch. Th factiv idx ca b xpadd i a Foui sis: ( ) = + mπz z m cos. m= Λ (-67) z x θ θ d i θ θ si(θ) Λ Fig..3. Diffactio of a moochomatic pla wav by a optically thic gatig. i = d =. Th compot of momtum ppdicula to th gatig vcto is cosvd. Th simplst cas is a siusoidal gatig, wh oly o-zo: ad th fist Foui compot a

58 39 πz ( z) = + cos. Λ (-68) Cosid th moochomatic wav diffactd fom a piodic mdium: th idx modulatio is lumpd to a aay of plas spaatd by qual distac, as illustatd i Fig..3. Assumig that th umb of plas is ifiit, th flctios fom ths plas a spcula. Th path diffc fo ays flctd fom adjact plas is Λ siθ, wh θ is th agl btw th icidt bam ad th gatig plas. Th itfc is costuctiv wh th phas diffc fo ighboig flctd ays is a itg multipl of π, which lads to Bagg's law: Λ siθ = N λ (-69) wh N is a itg, ad is th avag factiv idx of th mdium. Th agl θ satisfyig (-69) is dfid as th Bagg agl. I spac, th gatig piod is pstd by th gatig vcto, wh = π/ Λ is ow as th gatig wav umb. Bagg's law bcoms + = N, ad N is a vcto i i th (o-dimsioal) cipocal lattic. Th compot of th momtum ppdicula to th gatig vcto is cosvd. I a thi gatig, as dpictd i Fig..4, th tasvs dimsio of th piodic mdium is lativly small compad to th bam siz ad/o th wavlgth. Du to th fiit siz of th gatig plas, th diffactio fom ach pla should b cosidd i additio to th spcula flctio. Costuctiv itfc also occus i dictios oth tha th spcula flctio dictio. Th coditio fo costuctiv itfc is: Λ siθ + Λsiθ = N( λ/ ). (-7)

59 4 A dimsiolss paamt (Kli-Coo paamt), Q πλl/ Λ is itoducd to qualitativly dfi thic o thi gatigs, wh L is th thicss of th gatig. Wh Q >>, th gatig is classifid as a thic gatig, ad wh Q <<, th gatig is classifid as a thi gatig. z θ θ' x Λ Fig..4. Diffactio of a moochomatic pla wav by a optically thi gatig. θ may ot b qual to θ..4. Coupld Wav Thoy [6] Coupld wav thoy was itoducd by Kogli i 969 to tat thic dilctic gatigs ( Q >> ) [7]. I this appoach, th factio idx pofil was assumd to b siusoidal ad th dilctic mdium is isotopic. Th light popagatig i th piodic mdium is assumd to b polaizd light ad th diffactd light is coupld with th icidt bam. Coupld wav thoy is ow fo its simplicity ad vsatility to thotically simulat gatig diffactio phoma. Lt us cosid th light popagatio i a o-dimsioal piodic mdium with a siusoidal idx modulatio dscibd by (-68). Assum: () pla wavs hav uifom amplituds; () th gatig xtds latally to ifiity ( Q >>, thic gatig); ad (3) th lctic filds of both th

60 4 icidt light ad th diffactd light a paalll to th gatig plas ad ppdicula to th gatig vcto, as i Fig..5. I oth wods, oly th tasvs lctic (TE) mod is discussd. Basd o ths th assumptios, th lctic fild of th icidt ad diffactd bams ca b witt as: [ i( t )] + A xp[ i( ωt )] E = A xp ω (-7) wh ad a th wav vctos, ad A a th amplituds of th lctic filds, ad A ω is th fqucy of th bams. Th wav vctos ad th fqucy of th light a coctd by: = = / c. Wh th x z pla is st as th pla of icidc, th wav vctos ω ca th b witt i Catsia coodiats: ( α,, ) ad ( α,, ). Thfo, th lctic fild is: = β [ i( ωt α x β z) ] + A xp[ i( ωt α x β )]. = β E = A xp z (-7) (a) Tasmissio gatig z (b) Rflctio gatig y x L θ θ L θ θ Fig..5. Diffactio by optically thic gatigs. (a) Tasmissio gatig; (b) Rflctio gatig.

61 4 Th lctic fild satisfis th wav quatio: ω ( + ) E =, (-73) c wh is th factiv idx of th mdium Coupld Wav Thoy fo Tasmissio Gatigs Fo a tasmissio gatig with ifiit dimsio i th z dictio, th amplituds of th lctic filds a fuctios of x oly. I a mdium with a siusoidal idx modulatio, as dscibd i (-68), th wav quatio of th lctic fild ca b pstd by: = d d iα j dx dx ω cos Kz c j=, ω c j=, Aj xp j=, A xp ( ωt α x β z) A xp[ i ]. j j j [ i( ωt α x β z) ] [ i( ωt α x β z) ] j j j j j (-74) Th tm is glctd bcaus is usually of th od of to 3. Assumig th gy itchag btw th tasmittd wav ad th flctd wav is slow (i.., A ad A a slowly vayig fuctios), th scod od divativs of ad A a glctd. (-74) bcoms: A da iα xp dx ω = c + A xp( iα x iβ z)]. ( iα x iβ z) iα xp( iα x iβ z) [ xp( ikz) + xp( ikz) ][ A xp( iα x iβ z) da dx (-75) By multiplyig (-75) with xp( iα x iβ z), with j = o, ad itgatig ov z, two fist od difftial quatios a obtaid: j j

62 43 da dx da dx = = iκ A iκ A iδαx iδαx,, (-76) wh: κ π =, κ λ cosθ π Δα = α α = λ π =, λ cosθ ( cosθ cosθ ), (-77) ad th followig coditio is satisfid: β β ±. (-78) = K (a) Δα =, phas matchd z (b) Δα, phas mismatch z Δα K K x x Fig..6. K-spac of tasmissio gatigs (a) Idal phas match Δ α = ; (b) ot idal phas match Δα. Th phas mismatch Δ α dtmis th couplig ad gy xchag btw th bams. Wh cosθ = cosθ, Δ α =, th is a idal phas match, as show i Fig..6. Th tivial solutio θ = θ cospods to th tasmittd bam; th o-tivial solutio θ = θ cospods to th diffactd bam. Th Bagg coditio is satisfid wh Δ α = :

63 44 λ θ = θ = ± acsi θ B, (-79) Λ wh θ B is ow as th Bagg agl, ad th wav quatio bcoms: da dx da iκa, = iκ, (-8) dx = A π wh: κ =. (-8) λ cosθ Th solutios to (-8) a: A A A ( ) ( ) B ( x) = A ( ) cosκx ia ( ) siκx, ( x) = A ( ) cosκx ia ( ) siκx, wh ad a th lctic fild amplituds of th icidt bam ad th diffactd A bam at th icidt sufac. Fo a sigl icidt bam ( A A ( ) = ), th solutios bcom: ( x) A ( ) cos x, A ( x) = ia ( ) siκ. = x κ (-8) Egy is cosvd i this solutio. Th diffactio fficicy of th gatig is: ( L) ( ) I diffactd A η = = = si κl, (-83) I A icidt wh L is th thicss of th cll. Fo th idal gatigs i which all of ou assumptios a satisfid, th diffactio fficicy achs its fist maximum wh κl = π/. Wh κl > π/ (ov-modulatio), th diffactio fficicy dcass.

64 45 z Δα z Δα K K K x x (a) (b) Fig..7. Phas mismatch i tasmissio gatigs du to (a) agula dviatio; ad (b) wavlgth dviatio. Wh th icidt agl dviats slightly fom th Bagg agl, θ = θ B + Δθ, th diffactio agl dtmid by (-78) is: θ = θ B + Δθ. (-84) Th dviatio fom th Bagg agl sults i a phas mismatch giv by: Δα = Δθ siθ B = KΔθ, (-85) wh = π / λ, as illustatd i Fig..7(a). Fo th mismatchd cas, solutios to th coupld wav quatio (-76) a: Δα Δα A = s ( x) A ( ) xp i x i si sx + cos sx, Δα κ A sx s ( x) = ia ( ) xp i x si, Δα wh s = κ +. (-86)

65 46 Th cospodig diffactio fficicy is: η = A A ( L) () cosθ cosθ = κ Δα κ + Δα si κl + κ /. (-87) Th diffactio fficicy dcass with a icas of phas mismatch, as illustatd i Fig..8. Th agula dpdc of th diffactio fficicy ca b divd fom (-87). Th diffactio fficicy dops to / wh th dviatio of th icidt agl fom th Bagg agl is: Δθ = Δθ/ κ κλ = =. K π (-88) Diffactio Efficicy Phas Mismatch Fig..8. Th diffactio fficicy dcass with th icas of phas mismatch. Rd: L=π/3, Blac: L=π/. Phas Mismatch: Δα/. A dviatio i wavlgth, phas mismatch is giv by: Δ λ, ca also lad to a phas mismatch, as show i Fig..7 (b). Th

66 47 π Δλ Δ α =. (-89) Λ Λ cosθ B Th diffactio fficicy dcass to half th maximum valu wh th wavlgth of icidt bam chags by: Thfo, κλ Δ λ = Λ cosθ B. (-9) π Δ λ ca pst th spctal badwidth. Fig..9 plots th diffactio fficicy as a fuctio of th phas mismatch fo difft cll thicsss. Th abov discussio is valid fo TE wavs. Fo mo gal cass, th lctic fild is a vcto fild, ad th dilctic costat is a tso fild. Th piodic mdium is dscibd by ε ε + ε cos (-9) = ( ) wh ε is th avag dilctic tso, ad ε is th tso pstig th amplitud of th piodic dilctic modulatio. Followig th sam pocdus, th coupld wav quatios ca b divd, which hav th sam fom as thos of th TE wav (-76), but with difft couplig costats: κ κ = = π p λ cosθ ε π p λ cosθ ε ε p, ε p ; (-9) ad Δ α is giv by: Δα α α ± K. (-93) = x K is th x compot of th gatig vcto ; ad ad a th factiv idics x associatd with p s xp[ i( ω t )] ad p [ i( t )] xp ω, spctivly.

67 48 Diffactio Efficicy Phas Mismatch L=. L=.5 l=. L=.5 Fig..9. Diffactio fficicy as a fuctio of phas mismatch (Δβ/) fo flctio gatigs with difft thicsss L. Th diffactio fficicy is obtaid followig th sam appoach fo TE wavs. I th cas of TM wavs, th couplig costat is giv by κ p = κcosθ, wh Θ is th agl btw th lctic filds of th icidt bam ad th diffactd bam [7]. Th diffactio fficicy of th TM wav has th sam fom as th TE wav, but with a small couplig costat Coupld Wav Thoy fo Rflctio Gatigs Fo flctio gatigs, th bouday coditios qui that α = α (i.., θ = θ ), ad that th amplitud of both th icidt ad flctd bams b fuctios of z oly. Th coupld wav quatio fo flctio gatigs is dducd by followig a simila appoach to that of tasmissio gatigs:

68 49 da dz da dz = = iκ A iκ A iδ β z iδ β z (-94) = K wh: Δβ β β ±, (-95) π ad: κ =. λ siθ (-96) Th diffactio fficicy of a flctio gatig is divd by solvig (-94). I th cas of a pfct phas match ( Δ β = ), th diffactio fficicy is: η = tah κl. (-97) Simila to th cas of tasmissio gatigs, th couplig costat of th TM wav i a flctio gatig is giv by κ p = κcosθ, wh Θ is th agl btw th lctic filds of th icidt bam ad th diffactd bam. Th diffactio fficicy of th TM wav is always lss tha that of th TE wav fo flctio gatigs. Dviatio fom th Bagg coditio sults i a phas mismatch ( Δβ ) giv by: π 4π Δ β = siθ, (-98) Λ λ wh θ >, th diffactio fficicy of a flctio gatig with a phas mismatch is: κ sih sl η =, (-99) Δβ s cosh sl + sih sl Δβ wh: s = κ. (-) Diffactio fficicy as a fuctio of phas mismatch is illustatd i Fig...

69 5 Diffactio Efficicy L Fig... Diffactio fficicy of a flctio gatig as a fuctio of κ L fo a idally matchd phas Δβ=. Fom (-97) ad (-99), w otic that ov modulatio will ot occu i flctio gatigs. Fig.. modls a flctiv HPDLC gatig. Th piod of th gatig is m ad th thicss of th gatig is 5 μm. I 4x4 modlig, th liquid cystal lay ad th polym lay a assumd to b pu, ad thi factiv idics a.58 ad.5, spctivly. I th coupld wav thoy calculatio, th factiv idics a assumd to hav a si fuctio modulatio as dscibd i (-68) with =.55, =. 3.

70 5 Modlig of Rflctio Gatig Rflctio Wavlgth ( m ) 4x4 modlig coupld wav thoy Fig... Modlig of flctio gatig at omal icidc. Bma s 4x4 mthod ad coupld wav thoy calculatio a compad..5 Summay I this chapt, w hav ivstigatd light popagatio i statifid matials. Basd o Maxwll s quatios, both th Jos matix ad Bma s 4x4 mthod a discussd ad modlig was giv to xplai th gatig ffct of a cholstic ad HPDLC clls. Futhmo, coupld wav thoy was discussd with a focus o xplaiig diffactio by o-dimsioal tasmissio ad flctio mod phas gatigs. Th ffcts of vaious factos o diffactio fficicy w mathmatically ivstigatd. Whil oly isotopic gatigs a discussd i dtail, th aisotopic gatigs ca also b tatd with a simila mthod. Th sults of this chapt ca b applid to th HPDLC gatigs, which ca b appoximatd as phas gatigs.

71 CHAPTER 3 Hologaphic Polym Dispsd Liquid Cystal 3. Itoductio to Hologaphy I covtioal photogaphy, oly th distibutio of light itsity of th objct is codd ad costuctd. As atu of light ad its applicatio a futh sachd, th ida of codig th complt wav fild, both th phas ad th amplitud of light, is ispid. Rcostuctig th sam wav fot fom a hologam poducs th imag of th objct. Sic most codig mdia spod oly to th itsity of light, th phas ifomatio is tasfomd ito itsity duig a hologaphic codig by th itfc of moochomatic coht light. So, i dfiitio, a hologam is th codig of th itfc patt of multipl bams wh th codig mdium is positiod. Th itfc of coht light ays ca b dscibd by th suppositio of th lctic filds: N = E j j= E iωt. Th hologaphic codig tim is usually much log tha th piod of th light. Th avag light itsity ov a piod is I N N ( ) l ( ) m( ), * ( ) E( ) E ( ) = E E wh ( ) E l = m= I is th spatial distibutio of th light itsity, ( ) th spatial distibutio of th lctic fild, ad ( ) ad ( ) E l E m a th lctic fild compots of th difft codig bams. Th fist hologams w dmostatd by Gabo i 948 [8,9]. I Gabo's i-li hologaphy stup, th itfc patt was gatd by both th tasmittd bam passig though th objct ad th scattd light, as show i Fig. 3.. Gabo s hologaphy stup is oly 5

72 53 applicabl to objcts with high tasmittac ad th hologam is usually wa; two supimposd, out-of-focus twi imags could b obsvd. objct vitual imag hologam al imag poit souc codig mdium (a) (b) Fig. 3. Optical stup fo codig (a) ad costuctig (b) i-li hologam. objct bam vitual imag al imag hologam codig mdium fc bam (a) fc bam (b) Fig. 3.. Th optical stup fo codig (a) ad costuctig (b) off-li hologam.

73 54 Off-axis hologaphy was itoducd by Lith ad Upatis i 96 [], as illustatd i Fig. 3.. I off-axis hologaphy, th fc bam is spaatd fom th illumiatio bam ad is dictd to th photogaphic plat at a agl with th objct bam. Wh costuctd, th al imag ad th vitual imag a both locatd off th dict tasmittd bam ad a widly spaatd fom ach oth. Th imag quality of th hologam ca b impovd by adjustig th itsity of both bams to icas th cotast atio i th itfc patt. 3. Itoductio to Hologaphic Polym Dispsd Liquid Cystals Hologaphic polym dispsd liquid cystals (HPDLCs) a th succssos of taditioal polym dispsd liquid cystals (PDLC) that a compisd of a igid polym matix, ad adomly dispsd liquid cystal doplts isid th polym matix. Usually HPDLCs hav a statifid stuctu with altatig lays of liquid cystal ich ad polym ich plas, fomd by th hologaphic xposu of a mixtu of liquid cystals ad photo-polymizabl mooms. HPDLCs a omally tatd as switchabl phas gatigs. Th fist switchabl gatig basd o a PDLC was itoducd by Magum ad co-wos at Hughs Rsach Labs i 988 [ ]. A piodically modulatd distibutio of th liquid cystal doplt dsity i th polym two was gatd ud a masd UV xposu of th UV-cuabl PDLC mixtu, th piodic idx modulatio i th PDLC film fuctiod as a tasmissio gatig that could b dactivatd by th aligmt of th liquid cystal molculs i th LC doplts ud th xtal lctical fild. Th fist switchabl PDLC tasmissio gatig fomd by hologaphic xposu was dvlopd by Suthlad ad co-wos i 993 []. Th ppolym mixtu was cud i th visibl wavlgth ag. Th fist HPDLC flctio gatig was potd by Taaa ad cowos [3]. HPDLCs hav foud applicatios i displays [4, 5, 6] ad optical tlcommuicatio [7] with thi switchability ad thi lativly simpl fabicatio pocss.

74 55 To fabicat a covtioal HPDLC cll, two ITO glass substats with ati-flctio (AR) coatigs a assmbld with uifoly spad spacs to cotol th cll gap. Th codig mdium, o PDLC mixtu is sadwichd btw ths ITO glass substats. Th cll is xposd to th itfc patt gatd by two moochomatic coht las bams gally cayig qual pows ad polaizatios. Th light iducs a phas spaatio duig th photo polymizatio of th mooms i th mixtu ad causs th liquid cystals to b phas spaatd ito adomly oitd doplts i this pocss. O th oth had, as th lights cat gios of complt costuctiv ad dstuctiv itfc, th photo polymizatio i th bight (costuctiv) aa is fast tha i th da (dstuctiv) gio; th liquid cystals diffus to th da gio ad th mooms to th bight gio. Fially, a spatially piodic distibutio of polym ad liquid cystal (LC) doplts is gatd. Th avag factiv idx of th LC doplts, thus th factiv idx diffc btw th LC ich gio ad polym ich gio, ca b chagd with a aligmt of th liquid cystal molculs by a xtal lctic fild. I a omal mod HPDLC, factiv idics of th polym ad th LC doplt a umatchd without a xtal fild, whil i th vs mod, th factiv idics of th polym ad th LC doplts a matchd without a xtal fild. I ith cas, a xtal lctic fild ca switch th HPDLC fom a hologaphic stat to a o-hologaphic stat ad Vic Vsa. Th itfc of two collimatd las bams is utilizd to gat th itfc patt fo th hologaphic xposu. Th itfc patt of th two coht moochomatic bams has th fom: I ( ) = I + I + I I cosθ [( ) + φ ] cos (3-) wh ad I a th bam itsitis; θ is th agl btw th polaizatio dictio of th I codig bams; ad a th wav vctos of th codig bams; ad φ is th phas

75 56 diffc of th two bams. Th cotast atio of th itfc patt ca b dscibd by th fig costat: V cosθ II = I + I (3-) Wh I = I ad both codig bams a s-polaizd, th fig costat V is maximizd. Basd o th popagatio dictio of th diffactd light, covtioal HPDLCs ca b catgoizd ito two typs: () flctiv HPDLCs, as show i Fig. 3.3(a), wh th diffactd light popagats to th sam sid of th icidc, ad () tasmissiv HPDLCs, wh th diffactd light popagats to th opposit sid of th tasmittd light, as show i Fig. 3.3(b). Fig Hologaphic cofiguatios fo (a) flctiv mod ad (b) tasmissiv mod HPDLCs. If th codig bams a i th sam pla as th two witig bams, as illustatd i Fig. [ + ], wh 3.3, th itfc patt ca b pstd by: I = I V cos( Λx) Λ λ = siθ is th spatial piod (pitch) of th itfc patt, o th pitch of th HPDLC, ad th x axis is ppdicula to th HPDLC lays.

76 Tasmissio Mod HPDLCs Th covtioal tasmissio mod HPDLCs hav gatig plas oitd ppdicula to th cll sufacs. Th itfc patt is gatd by two coht las bams icidt fom th sam sid of th cll, as illustatd i Fig. 3.3(b); a statifid stuctu of polym ich ad liquid cystal ich lays is fomd by th xposu. Th pitch of th gatig is dtmid by: λ siθ w Λ = (3-3) w w wh θ w is th icidt agl isid th HPDLC, λ w is th codig bam wavlgth, ad w is th factiv idx at λ w. Th wavlgth flctd by th gatig is: λ λ cosθ w i = Λ cosθi = = Λsiθ = w cosθwi λw siθ siθ w w wh is th factiv idx of th HPDLC at th adig bam wavlgth λ, θ i is th adig bam icidt agl isid th HPDLC, ad θ is th agl btw th icidt (o diffactd) bam ad th gatig plas. Th diffactio poptis highly dpd o th diffc btw th factiv idx of th polym ich lay ad th liquid cystal ich lay. Fig. 3.4 is a schmatic illustatio of th opatio picipl of tasmissio mod HPDLCs. I th fild-off stat, light is stogly diffactd by th gatig; whil i th fild-o stat, light passs though as th gatig is dactivatd by th lctic fild. Th polaizatio dpdc of th diffactio fficicy of tasmissio gatigs was fist vald by Suthlad ad co-wos. Thi sach showd that th couplig cofficit fo th TE wav (s-polaizatio) ad th TM wav (p-polaizatio) a difft ad th idx modulatio of th TM wav was lag tha that of th TE wav [], which idicatd that th oitatio of LC doplts has a pfd dictio ppdicula to th gatig plas.

77 58 V V idx matchig lay LC ich AR coatig polym ich (a) (b) Fig Tasmissio HPDLCs i th (a) o stat; (b) off stat. Som HPDLC tasmissio gatigs with liquid cystals highly aligd alog spcific dictios w futh dvlopd by Vadaya ad co-wos [8]. Polym scaffoldigs fomd i a icomplt phas spaatio itcoct th ighboig polym ich lays ad povid a stog aligmt fo th liquid cystal molculs i th liquid cystal ich lays. A ffctiv polym fild is usd to mathmatically xplai th ffct of this aligmt, which is sposibl fo th lcto-optic spos ad optical aisotopy of th HPDLCs. Th dilctic aisotopy of th LC doplts i tasmissio mod HPDLCs was ivstigatd by Jazbis ad co-wos [9]. Ellipsoid shapd doplts i th LC ich pla was foud though SEM. Th idx modulatio is vald to b high alog th gatig vcto dictio, ad th dilctic aisotopy high i shot pitch HPDLCs, by fittig th masumt data of diffactio fficicy to th coupld wav thoy.

78 59 (a) p s G PR G s p Fig Polaizatio idpdt lcto-optical dvic basd o stacig two polaizatio ssitiv tasmissio mod HPDLCs (G ad G) ad a polaizatio otato (PR). Coutsy of Boio ad co-wos [3]. A polaizatio issitiv HPDLC dvic was dvlopd by Boio ad co-wos [3] usig two idtical tasmissio HPDLCs with a polaizatio otato i btw, as illustatd i Fig Th polaizatio otato fabicatd fom activ msog film ca otat th polaizatio of ctai wavlgth of light by π /, thus th s-polaizd light tavlig though th fist HPDLC is tasfomd to p-polaizd light by th polaizatio otato ad is th diffactd by th scod HPDLC, whil th p-polaizd light diffactd by th fist HPDLC is tasfomd to s-polaizd light ad th passd though th scod HPDLC. Th cotast atio of th diffactio fficicis of p-polaizatio ad s-polaizatio of th sigl tasmissio HPDLC i this dvic is high (~3) at 55 m. Th diffactio fficicy of th stacd dvic is as high as 98% fo o-polaizd light. A SEM photogaph of a tasmissio mod HPDLC opatig at ~ 5 m is illustatd i Fig. 3.6.

79 6 Fig Th SEM photogaph of a tasmissio mod HPDLC opatig at 5 m. Th top substat is pald bfo th SEM imag is captud. 3.4 Rflctio Mod HPDLC Rflctio mod HPDLCs a codd by las bams icidt fom difft sids of th cll, as illustatd i Fig. 3.3(a). Wh th two codig bams a symmtic, th piodic itfc patt is alog th cll omal dictio ad th sultig gatig plas a paalll to th cll sufacs. Uli th tasmissio mod HPDLCs, th LC doplts i flctiv HPDLCs a assumd to b adomly oitd i th pla paalll to th cll sufac, sic omally o polaizatio dpdc of flctio fficicy is idtifid i flctiv HPDLCs. Cosqutly, th avag factiv idx of th LC doplts is giv by: LC = o + 3 (3-4)

80 6 Fig Th SEM photogaph of a flctiv HPDLC opatig at ~5 m. Th imag is of th coss sctio of th cll. Howv, as th sult of a icomplt phas spaatio, th a still liquid cystals i th polym ich lay, ad th polym twos xtds ito th LC ich lay. Thfo, th avag factiv idx of both th polym ich gio,, ad th LC ich gio,, a i p LC th ag dfid by ad (th factiv idx of th polym). Th idx modulatio LC p LC p is lss tha LC p, ad is typically i th od of. Fo liquid cystals with a positiv dilctic aisotopy ( Δ ε > ), wh ad th LC o p molculs a aligd paalll to th applid lctic fild, th HPDLC cll will appa taspat to th adig light. Fig. 3.7 is a SEM photogaph of th polym two of a flctiv HPDLC opatig at aoud 5 m. Th SEM imag was ta aft th liquid cystals w washd away. Th statifid polym two ad cavitis that usd to b occupid by th liquid cystal a claly idtifid.

81 6 3.5 Vaiabl-Wavlgth HPDLC So fa most HPDLC latd lcto-optical dvics fuctio as a lctically switchabl gatig with two stats; th gatig stat (o switch-off stat) without a xtal lctic fild ad taspat stat (o switch-o stat) with a applid xtal lctic fild. Rctly, a vaiablwavlgth switchabl Bagg gatig fomd i polym-dispsd liquid cystals was pstd by th Display ad Photoics Laboatoy of Bow Uivsity [3]. This iovativ dvic ca switch btw two distictly difft flctig stats. A bldd moom systm was ppad by mixig Ebcyl 4866 with Ebcyl 83 (both fom UCB Radcu) i a atio of :. This was th mixd with th liquid cystal BL38 (EM Idustis) with a wight atio of 5:36:4 fo th mooms: liquid cystal : photoiitiato solutios, spctivly. This was homogizd ad mixd with a Tgito Mi-Foam X sufactat fom Uio Cabid(3 wt%). Th mixtu was ssitizd with a Ros Bgal ad N- Phylglyci i -viyl--pyolido photoiitiato so that th polymizatio could b caid out i th visibl wavlgth ag with a A las. A cll was fomd by dop fillig this mixtu btw ITO-coatd glasss. 5 μm spacs w usd to cotol th cll gap. + I th zo fild stat (Fig. 3.8(a)), th avag idx of th liquid cystal lay l ( E) is gat tha that of th polym lay ad a flctio pa is obsvd sic < ( E l ). As l l th fild is icasd as show i Fig. 3.8(b), th patial aligmt of th liquid cystal doplt achs a coditio wh ~ ( E l ), ad o flctio is obsvd bcaus th lays a l idx matchd ad optically homogous. At high lctic filds, as show i Fig. 3.8 (c), th liquid cystal bcoms highly aligd ad ( ) dcass to a valu blow, th sampl l E l flcts agai at a difft wavlgth. Fig. 3.9(a) shows a plot of pa wavlgth ad flctac as a fuctio of applid voltag. Rflctac is omalizd to th zo-voltag valu.

82 63 Th wavlgth spa is fom a zo-voltag valu of 45 m to a miimum of 438 m i th switchd stat. Fig. 3.9(b) shows th xpimtally masud flctac spctum at th zovoltag stat, th idx-matchd stat ( V), ad th switchd stat ( V). Fig Schmatic illustatio of a flctig vaiabl-wavlgth HPDLC. (a) Th avag idx of th liquid cystal lay ( ) is gat tha that of th polym lay ; (b) As th fild is applid, ~ ( E l ), ad th idx gatig is asd ad th sampl is optically homogous; (c) Th futh icasd fild gats th highly aligd stat wh ( E l ) dcass to a valu blow l l factio. Coutsy of C. C. Bowly t. al. [3]. l l E l. Th da lays cospod to lays of high idx of

83 64 (a) (b) Fig (a) Rflctac ad pa flctd wavlgth as a fuctio of applid voltag fo vaiabl wavlgth HPDLC with a 5 μm cll gap. (b) Expimtally (poits) ad modld (cuvs) flctac spcta of vaiabl wavlgth HPDLC masud at,, ad V. Cuvs w fit by vayig th ffctiv idx of th liquid cystal doplt ich plas ( E l ) usig a siusoidal idx pofil ad coupld wav thoy. Coutsy of C. C. Bowly t. al. [3].

84 HPDLC Matials Th HPDLC costitut matials usually cosist of liquid cystals, photo-polymizabl mooms/oligoms, ad a suitabl photoiitiato fo th xposu wavlgth. Sufactats may b addd to impov th itactio of th polym two ad th liquid cystal, i od to impov th lcto-optical pfomac ad th diffactio fficicy UV Mixtus Th fist HPDLCs w fabicatd usig UV cuabl mixtus [], adaptd ad dvlopd fom th PDLC matials basd o UV light-iduc polymizatio, ad a suitabl fo + hologaphic codig usig a 35 m A io las. A sis of fomulatios dvlopd fom PDLC mixtus cosists of PN393, which is a mixtu of low fuctioality acylat mooms ad photoiitiatos, ad o of th TL sis of liquid cystals (TL3, TL5, ad TL3) [3]. All ths matials a dvlopd fo PDLC applicatios by EMD Chmicals; th mixtu of PN393 ad TL liquid cystal is dsigd fo low itsity UV light cuig to fom PDLC with coss-lid polym twos. Tasmissio HPDLCs basd o this mixtu xhibit a polaizatio dpdc, ad th polym two is mchaically wa du to th low fuctioality of th PN393. A substitut matial fo PN393 was dvlopd by D Saa ad co-wos [33]. Thi fomula cosists of 8% -thylhxyl acylat (EHA), 5% Ebcyl 83, a hxafuctioal aliphatic utha acylat oligom (EB83), 5% timthylolpopa tiacylat (TMPTA), ad % UV photoiitiato DAROCUR 465 (Ciba Spcialtis), all i mass atio. Th w fomula is tmd MD393 ad has a fuctioality of ~.85. Th HPDLCs fabicatd usig th MD393 ad TL3 (mass atio :) hav high diffactio fficicis, high mchaical stability, ad low switchig voltags.

85 66 Th lcto-optical pfomac of HPDLC tasmissio gatigs ca b impovd by addig fluoiatd acylat, accodig to th sach sult of d Saa ad Qi [34]. Wh a factio of th moom mixtu MD393 is substitutd by th fluoiatd moom,,,-tifluoothyl acylat o,,,3,3,3-hxafluooisopopyl acylat, th diffactio fficicy icass, th switchig voltag dcass, ad th switchig tim iss with th icas of fluoiatd acylat coctatio Visibl Mixtus Th fist HPDLC mixtu cud by a visibl las was dvlopd by Suthlad ad cowos []. This HPDLC mixtu cotais th moom diptaythitol ptaacylat (DPHA), 3% liquid cystal E7, % of th coss-liig moom N-viylpyolido (NVP), 4 mols of photoiitiato Ros Bgal (RB), ad a small wight atio of co-iitiato N-phylglyci (NPG). Ros Bgal has a absoptio bad aoud 5~6 m, with a absoptio pa at 559 m [35]. Ud light xcitatio, Ros Bgal tasfs lctos to NPG gatig NPG adicals, which iitiat th f adical polymizatio. Th polymizatio pocss ca b dscibd i th followig quatios: RB + hv RB* RB* + NPG RB* + NPG NPG + M M NVP svs as a solvt fo th Ros Bgal ad NPG, hlpig th LC to dissolvi i th moom, ad also fuctios as a chai tmiato. Th mixtu of Ros Bgal, NPG, ad NVP ca b ppad spaatly as th photo iitiato solutio (P.I. solutio) bfo big addd to th moom-lc mixtu. Suthlad's mixtu allows fo th us of may covit hologaphic-

86 67 quality lass, at wavlgths icludig 488 m, 54.5 m ad 53 m. O dawbac of this mixtu is that th switchig voltag of th HPDLCs fabicatd with this mixtu is vy high. I Suthlad's fomula, a sufactat, viyl ooaoat (VN), ca b addd to th mixtu to duc th switchig voltag, ad th liquid cystal E7 ca b substitutd by BL38 with a high bifigc ( Δ =.7 at 589 m). Th w fomula cosists of BL38, DPHA, ad a P.I. solutio cosistig of 86% NVP, % NPG, ad 4% Ros Bgal. Th mass atio of th P.I. solutio is ~ 5% of th whol mixtu. Th optimizd atio of th liquid cystal BL38 ad moom DPHA ds to b dtmid xpimtally dpdig o th woig wavlgth of th HPDLC, ad diffactio agl, ad oth paamts. A HPDLC mixtu fo visibl light cuig cosistig of th LC BL38, th P.I. solutio, ad aliphatic utha si oligoms Ebcyl 83 (hxa-fuctioal) ad Ebcyl 4866 (tifuctioal) was dvlopd by Bowly ad co-wos [36]. Both Ebcyl 83 ad Ebcyl 4866 a mad fom UCB Radcu as th moom bld. Wh th fuctioality of th moom bld is ~4.5 with a mass atio ~ : of th two mooms, a maximum flctac of 7% was achivd Summay Th fudamtal cocpts ad opatioal picipls of HPDLC w discussd i this chapt, icludig th fabicatio ad opatio of both tasmissio mod ad flctiv HPDLCs. Th wavlgth vaiabl HPDLC was discussd i dtail. Th HPDLC matials fo both visibl ad UV cuig w itoducd.

87 CHAPTER 4 Liquid Cystal Faby-Pot 4. Itoductio I this chapt, w will discuss th fabicatio ad chaactizatio of liquid cystal Faby- Pot poducts fo applicatio i both spctal imagig ad optical tlcommuicatios. 4. Itoductio to Faby-Pot Itfomt Th Faby-Pot itfomt was dsigd by C. Faby ad A. Pot i 899 [37]. Th dvic cotais two patially flctig pla sufacs btw which ays of light fom multipl flctios cat itfc patts. W will bgi with a discussio of multipl-bam figs with a pla paalll plat, to illustat th picipl of a Faby-Pot itfomt. Fig. 4. is a diagam of a plat with factiv idx immsd i a bouday mdium with factiv idx of '. Th thicss of th plat is d. Suppos th flctio cofficits of light flctd by th two bouday a ad ', ad th tasmissio cofficit of light passig though th two boudais a t ad t', th complx amplituds of th wavs flctd fom th plat a [38] : (i) A, ' ' ( i ) iδ 3 ( tt A, ' ' i ) i δ ( p 3) ( i) i( p ) δ tt A, tt' ' A, wh (i) A is th amplitud of th icidt bam. δ is th phas diffc btw two ighboig bams that a flctd fom th plat 4πd cos( θ ) δ = (4-) λ 68

88 69 wh λ is th wavlgth i vacuum. Similaly, th complx amplituds of th wav tasmittd though th plat a, apat fom a uimpotat costat phas facto, (i) ( ) tt ' A, ' ' i i δ 4 ( ) tt A, ' ' i i δ ( p ) ( i) i( p ) δ tt A, tt' ' A, ' d ' Fig. 4. Diagam of a plat with a factio idx immsd i a bouday mdia with factio idx '. Fo ith polaizd compot, w hav tt ' = T, = - ', ad ' = = R, (4-) wh T is th tasmissivity ad R is th flctivity; thy a latd by R+T = accodig to gy cosvatio. If th fist p flctd wavs a supposd, th amplitud ( ) A ( p) of th lctic vcto of th flctd light is giv by th xpssio: A ( ) ( p) = { + tt' ' iδ ( + ' iδ ' ( p ) i( p ) δ )} A ( i) ( p ) ' = { + ( ' i( p ) δ iδ ) tt' ' iδ } A ( i) (4-3)

89 7 If th plat is sufficitly log, ad as P, w hav ) ( ' } ') ' ( '{ ) ( ) ( ) ( i tt A A A i i δ δ + = = (4-4) Cosidig (4-), w fid: ) ( R } ) ( ) ( ) ( ) ( i R A A A i i δ δ = = (4-5) So that th itsity of th flctd light is: ) ( cos ) cos ( )* ( ) ( ) ( i R R R I A A I δ δ + = = ) ( si 4 ) ( ) ( si 4 i R R R I δ δ + = (4-6) I a simila way, w obtai ) ( ' ' ) ( ) ( ) ( i tt t t A A A iδ = = (4-7) ad usig (4-), w hav ) ( R ) ( i T t A A iδ = (4-8) Th cospodig itsity of th tasmittd light is: ) ( cos )* ( ) ( ) ( i R R T t t t I A A I δ + = = ) ( si 4 ) ( i R R T I δ + = (4-9) (4-6) ad (4-9) a ow as Aiy s fomula. Wh a paamt F, is dfid as: ) ( 4 R R F = (4-) Th itsity distibutios of th flctd ad tasmittd patts a giv by: ) / ( si ) / ( si ) ( ) ( δ δ F F I I i + = (4-)

90 7 I I ( t) ( i) = + F si ( δ / ) (4-) Evidtly th two patts a complmtay, i th ss that I I ( ) ( i) I I ( t) + ( i ) = (4-3) Equatio (4-3) shows gy is cosvd wh th mdium has o absoptio. Wh δ=mπ, ( t) ( i) ( t) ( i) I / I has a maximum valu; wh δ = π ( m + ), I / I has a miimizd valu. ( m is a itg ). Wh R is big, th patt i th tasmittd light cosists of aow bight figs o a almost compltly da bacgoud ad, similaly, th patt i th flctd light bcoms o of aow da figs o a othwis aly uifom bight bacgoud. Th shapss of th figs is covitly masud by thi half-itsity width. Th atio of th spaatio of th adjact figs ad th half-width is dfid as th fiss, Ғ, of th figs. Fo th fig of a itgal od m, th poits wh th itsity is half its maximum valu a at ε δ = πm ± (4-4) ad + F si ( ε / ) = (4-5) Wh F is sufficitly lag, ε is so small that w may assum si(ε/4)=ε/4 i (4-5); th halfwidth is obtaid to b: 4 ε = (4-6) F Th fiss is th Ғ = π F π R = = επ R (4-7)

91 7 Fig. 4. shows th bhavio of I ( t) / I ( i) as a fuctio of th phas diffc δ fo vaious valus of fiss Ғ. Tasmissio Pfomac of difft Fiss Numb Tasmissio Phas Shift Fiss=3 Fiss=5 Fiss= Fiss= Fig. 4. Bhavio of I ( t) / I ( i) as a fuctio of th phas diffc δ fo vaious valus of fiss Ғ. Uit of th phas diffc δ is π. Th so fa discussd multipl bam itfc figs fom a pla paalll plat ca also b applid to th ai-gap Faby-Pot itfomt wh th icidt light is at a omal icidc. A ai-gap Faby-Pot itfomt cosists of two glass o quatz plats P, P (Fig. 4.3) with pla sufacs. Th i sufacs a coatd with patially taspat films of high flctivity, ad a paalll, so that thy clos a pla paalll plat of ai with fixd spaatio, d, dcidd by a spac. This fom of th itfomt is oft fd to as Faby- Pot talo.

92 73 d P P Fig Faby-Pot itfomt Wh th icidt light is ot collimatd, th itsity of light I(θ,λ) tasmittd though a idal Faby-Pot talo (o with o dfcts) is giv by I θλ = λ o + with (, ) I ( ) ( / ) si F π ( δ /) 4π d cos( θ ) δ = (4-8) λ wh λ is th light s wavlgth, I o (λ) is th itsity i th ct of ach fig, d is th plat spaatio, is th idx of factio of th matial btw th talo plats. (fo a ai-gap, =.) ad F is th fiss of th talo dfid i (4-7). Fom ow o, w will us F to pst th fiss istad of Ғ. Wh th light souc is moochomatic, th Faby-Pot allows tasmittac of light at oly spcific icidc agls. Imagig th output of th Faby- Pot poducs a sis of cicula figs such as thos show i Fig As th wavlgth of th light dcass, th diamts of th igs icas util thy vtually occupy th spac lft vacat by th xt adjact xtal ig. As this happs, a w ig appas i th ct of th patt to plac th old o.

93 74 Fig Imag of th Faby-Pot itfc patt with moochomatic icidt light. Idally th fiss F is oly a fuctio of flctivity R, th flctiv fiss, if th talo plats a pfct with o sufac dfcts ad th two talo-plats a pfctly paalll. Fig. 4.5 shows th latio btw th flctiv fiss with th flctivity. I ality, v th bst talo will possss dfcts that limit th thotically xpctd pfomac. Th actual fiss will usually b low tha th flctiv fiss. If th gap is spcifid as d ± λ /5, w ca assum th gap will b withi ths limits fo most of its aa, say 95%. At ay poit th most pobabl valu of th gap thicss is d ad th umb of placs wh it dpats fom this by a lag amout will b vy small. Th total aa wh th gap dpats fom d by a amout btw x ad x+dx ca b covitly xpssd as th Gaussia fomula: ba da ( x) = xp( b x ) dx, wh b is a figu of mit which icass as th gap bcoms π mo ad mo uifom. Thus difft pats of th gap will cotibut to th total itsity at slightly difft ods of itfc ad th ffct vals itslf as a covolutio of th thotical Aiy pofil with a Gaussia cuv. If w assum that a a maximum th Aiy pofil is appoximatly th sam shap as a Gaussia cuv, ad mmb that th covolutio

94 75 of two Gaussias is aoth Gaussia with a half-itsity width which is th Pythagoa sum of th compot half-itsity widths, th w ca dtmi th flctio cofficit that th plats should hav, povidd that w ow th flatss of thm. If thy a flat to λ /5, th th bst possibl fiss is F=5. If th flctig lays hav a flctio such that th flctio fiss would b 5, th th ffctiv fiss would b 5 /, o about 35. Th typs of dfcts that cotibut to this ductio a sphical dfcts, sufac igulaitis, ad paalllism dfcts, as show i Fig Rflctiv Fiss Vs Rflctivity Rflctiv Fiss Rflctivity ( % ) Fig. 4.5 Rlatio of th flctiv fiss with th flctivity. (a) (b) (c) Fig Sphical dfcts (a), sufac igulaitis (b), ad paalllism dfcts (c).

95 76 Effctiv Fiss VS Dfct Fiss Effctiv Fiss Dfct Fiss FR=5 FR= FR=5 Fig Effctiv fiss chags with th dfct fiss. FR psts th flctiv fiss. Th latd dfct fiss a F, F, ad F. Th total dfct fiss F is dcidd by: ds dg dp d F d = F ds + F dg + F dp Th actual ffctiv fiss F is dcidd by: F = F d + F R fiss. Fig. 4.7 Plots how th ffctiv fiss chags with th dfct fiss fo difft flctiv To us th Faby-Pot talo as a filt, it is customay to stict th icidc agl of th light to match that of th imost ig, o spctal lmt. Th sultig fild-of-viw (FOV) is giv by 8δλ FOV = (4-9) λ

96 77 wh δλ is th spctal solutio of th talo. Th F Spctal Rag (FSR) is dfid as th wavlgth diffc of th two ighboig costuctiv itfc pas, o tasmissio pas, ad is chaactizd pdomiatly by th gap of th talo. Th coditio fo th costuctiv itfc is dtmid by: d m λ = (4-) Wh is th factio idx, d is th gap, λ is th wavlgth ad m is th od umb. Thus th FSR btw th m-th od ad th m+-th od is giv by: d FSR = mm ( + ) (4-) Nomally m is a lag umb, so it bcoms: λ FSR =Δ λ = (4-) d Fo th high-solutio talo, th f spctal ag is limitd by th afomtiod ffctiv fiss. I od to xpad th f spctal ag, two talos ca b usd i sis, th o with a lag gap, ow as th solvig talo, dfis th spctal solutio of th systm. Th talo with th smallst gap, ow as th suppssio talo, suppsss som of th ods of th solvig talo. Th sult is a systm with th spctal solutio of th solvig talo ad a FSR lag o qual to that of th suppssio talo. Th latt dpds o th atio of th FSR of th two talos with spct to ach oth. Suppos that th atio of th FSR of th suppssio talo to th solvig talo ca b xpssd as atio of itgs A/B, wh A ad B do ot hav a commo diviso. Th FSR of th twi-talo systm is th giv by

97 78 FSR = A FSRsol = B (4-3) FSR sup Fig Modlig of twi talo systm with th gaps of 3 mico ad mico. Wh FSR sol ad FSR sup a th f spctal ags of th solvig ad suppssio talos, spctivly. If B =, th suppssio talo dfis th FSR of th systm, othwis th FSR of th systm will b lag tha that of th suppssio talo by a facto of B. I th twi-talo systm, th FSR of th solvig talo has ow b xpadd by th facto A, ad so has th umb of availabl spctal solutio lmt. Th facto A will hcfoth b calld

98 79 th FSR xpasio facto. Fig. 4.8 modls twi talo systm, o with th gap of 3 mico ad th oth with a gap of mico. 4.3 Itoductio to Liquid Cystal Faby-Pot (LCFP) Tuabl Filt Tuabl filts with a wid tuabl ag that cov th whol C-bad, o L-bad, o both, hav foud wid applicatios i fib optical commuicatio systms, maily i th domais: tuabl lass, wavlgth divisio multiplxig systms (WDM), ad chal moitoig. Du to th gowig dmad i bit-ats ad th umb of chals i a WDM systm, tuabl aow bad pass filts a quid [39]. So fa, may ffots hav b mad to ma tuabl filts basd o th Faby-Pot picipl [4,4,4]. Th tuabl optical filts usig a LiNbO 3 tosioal actuato [4] d a high divig voltags up to 5 volts ad hav a aow tuabl ag. Th tuabl filt [4] potd by M. Iodic is a tmpatu-tud silico talo filt with a aow passbad; howv it posssss a aow f spctal ag ad th tuig spd was ot potd. Employig a liquid cystal matial as a cavity mdium i a Faby-Pot talo has may mits such as low divig voltag, low istio loss, ad wid tuig ag. Tuabl LCFPs w fist poposd by Mada i 99 [43] ad Patl i 99 [44], both fom Bllco. Hiabayashi t. al. fom NTT [45] ad Bao t. al. fom Coloado Uivsity [46] hav also wod o LCFP. Th fabicatio of a LCFP dvic is almost idtical to th fabicatio of a homogously aligd matic liquid cystal cll, o a ECB cll. Th talo plats a coatd with a coductiv ITO lays, ad th high flctio multi-lay dilctic coatigs. A polyimid lay with a thicss of about 5~ m is spi-coatd o o sid ad is th uifomly ubbd to gat th aligmt lay fo th liquid cystals. Th talo plats a th assmbld with spacs to

99 8 cotol th thicss of Faby-Pot cavity. Fially, th liquid cystal matial is vacuum-filld ito th cavity. Fig. 4.9 shows th stuctu of a liquid cystal Faby-Pot talo. High Rflctio dilctic coatig Liquid cystal Etalo Plat ITO lay Fig. 4.9 Stuctu of liquid cystal Faby-Pot. Th LCFP is polaizatio dpdt. Th factiv idx fo light with a polaizatio dictio paalll to th ubbig dictio, o -compot, is, ad th factiv idx fo light with a polaizatio dictio ppdicula to th ubbig dictio, o o-compot, is. LCFPs a tuabl oly fo -compots bcaus ca b tud with a applid voltag, thus a polaiz is cssay to b placd paalll to th aligmt dictio of th liquid cystal to allow th xtaodiay mod of light to pass though ad bloc th odiay mod. Wh voltag is applid, th factiv idx of th -compot chags alog th omal dictio of th cll with a chag of th dicto cofiguatio isid th matic cll, as was discussd i Chapt ; th avag factiv idx of th -compot isid th cavity is: o

100 8 = d d ( z) dz (4-4) Fig. 4. shows how th avag factiv idx chags with a applid voltag. Th modl is basd o th modlig of dicto cofiguatio dscibd i chapt. Th paamts usd fo th modl a: =.79, =.53, K = 5.5, K33 = 8., ε // =5. 5, ε = 5.. Th ptilt agl was assumd to b. o Avag Rfactio Idx Vs Voltag Avag Rfactio Idx Applid Voltag (v) Fig. 4.. Th avag factiv idx chags with th applid voltags. Fo ctai applicatios, it is dsiabl to hav a tuabl filt without a polaizatio dpdc, such as i fib optic tlcommuicatio wh th polaizatio stat of th optical sigal may b uow. Sval mthods hav b discussd to fabicat a polaizatioissitiv Faby-Pot dvic [5]. Fig. 4. shows a solutio to achiv th polaizatio

101 8 idpdt optical filtig ad tuig. Th icidt light (polaizatio stat uow) is spaatd ito two lialy polaizd bams by a polaizatio bam-splitt. Each bam passs though a LCFP with th liquid cystal aligmt dictio paalll to th polaizatio dictio, ad is wavlgth-filtd by th tuabl LCFP fil. Fially, a scod polaizatio bam splitt (ot show i th figu) will combi th two filtd bams bac ito o bam. Polaizatio Bam splitt LCFP Mio LCFP Fig. 4.. Combiatio of polaizatio bam splitt ad two LCFPs with aligmt dictios ppdicula to ach oth, to achiv th polaizatio-idpdt wavlgth filtig. Fig. 4. shows a aoth solutio fo a polaizatio idpdt optical filtig ad tuig systm. Isid th Faby-Pot cavity, th a two thicss idtical liquid cystal lays with th aligmt dictios ppdicula to ach oth. Bcaus of th symmty, light with ay polaizatio stat has a idtical optical path lgth of d + d t, wh is th o + g g factiv idx of th glass substat isid th cavity. Whil i this stuctu, wavlgth filtig is tuabl by chagig though a applid voltag; th tuabl ag is dcasd by

102 83 a facto of d, compad with LCFP with a sigl liquid cystal lay. Th stuctu d + d + t o g dscibd i Fig. 4. is much mo compact tha that i Fig. 4.; howv, as th glass thicss t is much lag tha th liquid cystal thicss d, th tuabl ag is vy small. Th twistd matic Faby-Pot itfomt (TN-FPI) was poposd by Patl ad L i 99 [47]. H th liquid cystal aligmt dictios at th two opposit talo plats a ppdicula to ach oth; thus, th liquid cystal lay has a 9 twist though th whol cavity, as show i Fig A TN-FPI wos i a high divig voltag gio, wh, i th middl of th cavity, th liquid cystals a homotopicly aligd du to th lctic fild, th two sidual homogous liquid cystal lays clos to th substat sufacs will compsat ach oth ad mas th wavlgth-filtig tuabl ad polaizatio idpdt. t d d Fig. 4.. Two LC lays isid a Faby-Pot cavity a usd to achiv a polaizatio idpdt wavlgth filtig ad tuig.

103 84 Etalo Plat High Rflctio dilctic coatig Twistd Liquid cystal ITO lay Polyimid aligmt lay Fig Tuabl ad polaizatio idpdt twistd matic Faby-Pot talo. 4.4 Fabicatio ad Tstig of LCFP Tuabl Filt Sigl LCFP Systm Tuabl LCFP itfomts hav b pov to b comptitiv i spctoscopy, LIDAR ad IR imagig [48,49,6,6]. Th advatags of a tuabl liquid cystal Faby-Pot filt a: high solutio, wid f spctal ag ad tuabl ag, fast spos tim, solid stat (o movig pats), low divig voltag, ad lag aptu. W fabicatd a LCFP (#68) fo applicatios i spctal imagig. Th talo plats usd had diamts of 38 mm ad sufac flatss of / at 63.8 m. A flctiv coatig with a 9% flctivity fom 7- m was dpositd o ach of th talo plats [6]. Th talo was filld with a -mico thic lay of matic liquid cystal ad gappd usig small sphical fusd silica spacs. Fial paalllism aligmt was mad usig a UVB cud adhsiv whil moitoig th fig patt.

104 85 Tasmissio cuvs as a fuctio of wavlgth w obtaid as show i Fig. 4.4 though Fig To ma ths masumts, light fom a moochomato is collimatd ad passd though a polaiz bfo achig th LCFP that is big tstd. Th bam is th focussd oto a photodiod dtcto ad moitod by a comput. Th spctal solutio of this st-up is about. m, ad th bam is collimatd such that oly th imost od of th LCFP is sampld. Th polaiz s aim is to mov th polaizatio compot of th bam that th LCFP caot tu. To obtai a tasmissio cuv, th moochomato is scad i wavlgth whil a spcific voltag is applid acoss th LCFP s gap. Th moochomato is th scad i wavlgth oc agai with th LCFP out of th bam. Dividig th fom by th latt, th tasmissio of th LCFP as a fuctio of wavlgth is obtaid. Th LCFP (#68) achivd a fiss of 9- dpdig o wavlgth ad applid voltag. Pa tasmissio of polaizd light agd fom 4-7%. Th tstig was pfomd o a full woig aptu, which is. ich i diamt, 8% of.5 ich. Th fiss ad f spctal ag (FSR) is summaizd i Tabl 4.. Th lctooptical spos masud at th wavlgth 85 m is show i Fig Fig. 4.8 shows a imag of LCFP #68 istalld i th housig with a lctical cocto. Fom Tabl 4., th td is appat that th fiss icass with th wavlgth. This is bcaus, whil th sufac quality of th talo plat is ot wavlgth dpdt i th uit of m, it is omally judgd by λ/m, wh λ is th woig wavlgth ad m is a umb, th dfct fiss ca b stimatd by: F d = m. This mas, suppos th dfct fiss at th 5 m is, th dfct fiss at th m is ~. Thus th total fiss icass with wavlgth supposig th flctivity (o flctiv fiss) is pt uchagd.

105 86 W hav also succssfully fabicatd a liquid cystal Faby-Pot fo optical tlcommuicatio applicatios. Tabl 4. shows th sult of a tuabl LCFP filt i th NIR ag (C-bad) dsigd fo a tuabl las applicatio. W hav succssfully modifid th highflctio coatig dsig to cov th woig ag fom 5 m to 57 m. W hav also optimizd th liquid cystal matial ad aligmt polyimid to miimiz th tasmissio loss. Th cavity gap is cotolld by μm spacs. Fiss ad F Spctal Rag of LCFP #68 Voltag Applid: -->.5 V 3.5 V 9 V Fiss at ~ 7 m FSR at ~7m 6.84 m 8.46m 9.4m Fiss at ~ 8 m FSR at ~8m 8.75m.4m.66m Fiss at ~ m FSR at ~m 6.3m 7.48m 8.39m Tabl 4.. Fiss ad f spctal ag of LCFP # 68 at difft voltags. Paamt Masud Valu Plat Diamt mm Ct Wavlgth 55m Tst Aa. squa-mm Istio Loss.5db F Spctal Rag 4 THz (37m) Spctal Rsolutio 8 GHz (. m) Fiss 3 Rspos tim ms Tabl 4.. Tstig sult of tuabl LCFP fo tuabl las i NIR ag.

106 87 Spctal Rspos of LCFP # 68 (.5 V voltag applid ) Tasmissio Wavlgth (A) Fig Spctal spos of LCFP #68, masud at.5 V. Th polaiz is paalll to th dicto dictio of th liquid cystals.

107 88 Spctal Rspos of LCFP # 68 (3.5 V voltag applid ) Tasmissio Wavlgth (A) Fig Spctal spos of LCFP #68, masud at 3.5 V. Th polaiz is paalll to th dicto dictio of th liquid cystals.

108 89 Spctal Rspos of LCFP # 68 (9 V voltag applid ) Tasmissio Wavlgth (A) Fig Spctal spos of LCFP #68, masud at 9. V. Th polaiz is paalll to th dicto dictio of th liquid cystals.

109 9 Voltag Sca of LCFP #68 ( at 85m ) Fig Elcto-optical spos of LCFP #68, masud at 85 m. Fig LCFP #68 i th housig with lctical cocto.

110 Twi LCFP Systm W fabicatd a twi-lcfp systm fo th applicatio of high-solutio spctal imagig. Two LCFP talos w costuctd. Etalo plats hav a diamt of 38 mm ad a sufac flatss of / at a wavlgth of 63.8 m. A flctiv coatig with a 9% flctivity fom 7- m was dpositd o ach of th talo plats. Th fist talo (solvig talo) was filld with a 3-mico thic lay of matic liquid cystal ad gappd usig small sphical fusd silica spacs. Fial paalllism aligmt was mad usig a UV cud adhsiv whil moitoig th fig patt. Fig Photogaphs of th sigl talo i th housig (ight) ad th twi talo imagig filt (lft). As th high flctio coatig xhibit a phas shift upo flctio that altd th ffctiv gap, th suppssio talo was costuctd with a 6 mico gap ath th a 7.5 mico gap i od to compsat fo th phas shift. Th talos a housd i a cylidical housig (Fig. 4.9) that cosists of a i cylid that holds th talo with uby spacs isolatig th talo fom th housig. Th scod cylid holds th hatig lmt ad thmostat. Usig a PID cotolld hatig systm th talo

111 9 tmpatu ca b maitaid to. dgs Clsius. Th housig has a dovtail flag that allows th talos to b coctd ad th otatd with spct to ach oth aoud th optical axis. This otatioal fatu allows fo th aligmt of th polaizatio stats of th talos to b co-icidt. Tasmissio Wavlgth (Agstom) Fig. 4.. Tasmissio as a fuctio of wavlgth fo th 3 μm gap LCFP.

112 93 Tasmissio Wavlgth (Agstom) Fig. 4.. Tasmissio as a fuctio of wavlgth fo th 6 μm gap LCFP. Fig. 4. shows th tasmissio vsus wavlgth with a applid pottial of 9 Volts, fo th LCFP with 3 μm spac. Fig. 4. shows th sam tstig fo th LCFP with 6.μm spacs. Both talos achivd a fiss of 9- dpdig o wavlgth ad applid voltag. Th pa tasmissio of polaizd light agd fom 4-7% Eviomt Tst of LCFP Two viomtal tsts w pfomd at th Utah Stat Uivsity Spac Dyamics Laboatoy o talos simila to thos dscibd abov. Th talos usd i this sis of tstig w of a small diamt (5 mm) ad had a multilay dilctic flcto with a 9% flctivity fom 5-7 m. A gap of micos was usd i ach talo. Th talos w

113 94 tstd usig th moochomt dscibd pviously ad aft th sha ad thmal-vacuum tsts. Th itt of this tst was to thmally cycl th LCFP btw at last 4 C ad C with a appoximat dwll tim of 6 miuts fo ach of th two tmpatu xtms ad to xpos th talos to a G sha quivalt to a lauch o a Pgasus lauch vhicl. Th tmpatu data fom th F-P Liquid Cystal thmal vacuum tst is show i Fig. 4.. Iitially th tmpatu cycls did ot achiv th quid tmpatu of C duig th cold cycls. Th tmpatu pogam was modifid to mt th C quimt, sultig i fiv accptabl thmal cycls. Th tmpatu cyclig at was 5 C/mi, with a dwll tim of 3 miuts at ach d. Th thmal coupl usd to moito th talo tmpatu was attachd to th sid of th housig that covd th LCFP. A cusoy xamiatio of th two LCFP talos did ot val ay obvious damag fom ith th sha tst o th thmal cyclig; th talo plats w ith chippd o cacd, ad th lctical wis w still solidly attachd to th substat. Tasmissio cuvs as a fuctio of wavlgth w obtaid fo ach of th two talos usig th tchiqu dscibd pviously. This was do fo two voltag sttigs of th LC-FP talos. As show i Figus 4.3 ad 4.4, th data sts had to b shiftd i wavlgth fo th pas to b lid up, which ca b attibutd to two thigs. Fist, uctaitis i th oitatio of th LCFP with spct to th bam ca caus a shift of th pa locatio ach tim a talo is st-up i th bam. Scodly, uctaitis i th absolut wavlgth calibatio cuvs could b a facto.

114 95 Thmal Vacuum Tmpatu Cyclig Rsults fo F-P Liquid Cystal icludig Plimiay Tst Cycls TC #4 attachd to th sid of F-P LC housig 5 Expadd Fial Tmpatu Cycls Show i "Fial Dwll Cycls" Chat. Accptabl Rsults Tampatu C TC#4 attachd to sid wall of housig Plimiay Tmpatu Cycl Tst Th F-P LC did ot ach th quid tmpatu of -C ad th dwll tim quid adjustmt. Uaccptabl tst. - 6//3 5:36 6//3 :48 6//3 6: 6//3 3: 6//3 :4 6/3/3 3:36 6/3/3 :48 Tim (m/d/y h:mi) Fig. 4.. Tmpatu vsus tim fo th thmal vacuum tstig of th LCFP

115 96 Fig Tasmissio of th LCFP that udwt a Pgasus-lvl sha tst fo two difft voltag sttigs ( ad 9 Volt). Th offst of th cuvs i tasmissio ad wavlgth is idicatd i pathsis.

116 97 Fig Tasmissio of th LCFP that udwt thmal cyclig, bfo ad aft th thmal cyclig fo two difft voltag sttigs ( ad 9 Volt). Th offst of th cuvs i tasmissio ad wavlgth is idicatd i pathsis.

117 Summay ad Coclusios W hav aalyzd, fabicatd ad chaactizd liquid cystal Faby-Pot poducts fo applicatio i both spctal imagig ad optical tlcommuicatio. Both sigl-talo ad twitalo systms w fabicatd. A Fiss of mo tha i th visibl wavlgth ag ad a fiss of mo tha 3 i NIR w achivd fo th tuabl LCFP poduct.

118 CHAPTER 5 Switchabl Cicl-to-Poit Covt 5. Itoductio This chapt discusss iovativ switchabl cicula-to-poit covt (SCPC) dvics basd o hologaphic polym dispsd liquid cystal (HPDLC) tchology, Faby-Pot itfomts, ad th hologaphic cicula-to-poit covt (HCPC). W will discuss th cocpt ad dsig of a iovativ SCPC dvic, ad th fabicatio ad chaactizatio of SCPC dvics woig at difft wavlgths (visibl ad NIR), ad with difft chal umbs (sigl chal, chal, ad 3 chal). 5. Bacgoud: Itoductio to HCPC Faby-Pot itfomts (FPI) a mployd as spctal-solvig lmts i vaious applicatios, such as i Lida dtctio of atomosphical, viomtal, ad climat chags [48,49] ad i tlcommuicatios [43,59]. I a dict dtctio Doppl Lida o icoht Lida systm to masu wid vlocitis by aosol ad/o molcula bacscatt, th Doppl shift sultig i a puls of aowbad las light fom scattig by aosols o molculs is masud. A fc spctum of a outgoig las bam is masud by th collctio of light scattd fom th zo-wid bacgoud. Wh th tu sigal of a bacscattd las light passs though th civig optics, th Doppl shift ca b dtmid by subtactig th fc spctum fom th tu sigal. A high solutio Faby-Pot itfomt is usd to dtct th wavlgth shifts. 99

119 Th Faby-Pot itfomt poducs a cicula itfc spctum o fig patts of qual aa igs pstig qual wavlgth itvals, shaig a commo axis, at th ifiity focus of a objctiv ls systm. Th is a log-stablishd poblm of collctig ad tstig th sigal fom th cicula fig patt as it s difficult o xpsiv to dsig dtctos with ig-shapd gomtis. Difft typs of imag pla dtctos hav b catd which attmpt to match th cicula patt. O such imag pla dtcto was potd by Timothy t al. i 983 [5]. Thi dvic cosistd of a S- photocathod, th mico-chal plat lcto multiplicatio stags, ad a qual-aa coctic-ig sgmtd aod to match th itfc ig patt. Aoth imag pla dtcto ivtd by Bissott t al. was a multi-lmt dtcto of coctic igs of PIN photodiod matial [5, 53]. All of ths imag pla dtctos typically suffd fom bluig of spot sizs ad low quatum fficicy. A difft appoach fo covtig th Faby-Pot fig patt itslf to fit lia dtctos has also b accomplishd [54, 55]. A 45 half agl itally flctig co sgmt is usd to covt th cicula Faby-Pot itfomt fig patt ito a lia patt. McGill ad co-wos dvlopd th passiv hologaphic optical lmt fo Lida dtctio [56,57,58]. Th hologaphic optical lmt compiss aas, ach of which acts as a spaat ls to imag th light icidt i its aa to a imag poit. Each aa cotais th codd hologam of a poit souc objct. Th imag poits ca b mad to li i a li i th sam focal pla so as to alig with a lia aay dtcto. Hologaphic Cicula-to-poit covt (HCPC) hav b dvlopd [57] that hav coctic qual aas to match th cicula fig patt of a Faby-Pot itfomt. A HCPC has a high tasmissio fficicy, ad wh coupld with a high quatum fficicy solid stat dtcto, povids a fficit photo-collctig dtctio systm fo a Faby-Pot itfomt. Th HCPC, as wll as oth hologaphic lmts, may

120 b usd as pat of th dtctio systm i a dict dtctio Doppl Lida systm o multipl fild of viw Lida systm. HCPC hologaphic plat is dividd ito coctic auli. Each aulus of th hologaphic plat fuctios as a sigl ls ad covgs th icidt bam to a poit focus, as dpictd i Fig. 5.. I od to match th Faby-Pot fig patt, th auli a dsigd to itcpt qual wavlgth itvals. Th sigals fom difft wavlgths a spatially discimiatd by th HCPC dvic, whas multipl dtctos a quid fo McGill's HCPC dvics. Fig. 5.. Th ay tac diagam of th hologaphic cicula-to-poit covt (HCPC) dvlopd by McGill ad co-wos. All light icidt oto a giv aulus of th HCPC is dictd to a dsigatd poit. Th focal poits appa i a commo focal pla paalll to th HCPC plat ad a agulaly spaatd.

121 5.3 Picipl of Opatio of SCPC Th switchabl cicl-to-poit covt (SCPC) is a combiatio of th HCPC ad th HPDLC that allows fo a lctic cotollability, whil maitaiig th taditioal highlights of a Faby-Pot itfomt: high optical thoughput ad high spctal solutio. Basd o HPDLC tchology, th SCPC dvic is dsigd to covt th sigal fom a Faby-Pot talo to a focus poit o a aay of poits just li th HCPC; what is mo, th covsio ca b dactivatd by applyig a stog ough lctic fild, o th covsio ca b lctically switchd o ad off [6]. Wh a Idium Ti Oxid (ITO) coductiv lay o o of th SCPC substats is pattd with cicula pixls that match th Faby-Pot cicula itfc patt, idividual chal a discimiatd by th Faby-Pot talo ad ca b spaatly slctd ad switchd o ad off. Fig. 5. is a coss sctioal dawig of a 4x switch mployig two idtical SCPC lmts i sis ach lmt has 4 ig-chals. Both of th SCPC lmts hav simila ig pixl patts gomtically matchig th Faby-Pot itfc ig patt, ad both a dsigd so that ach ig pixl covts th gy withi a cospodig cicula wavlgth chal to th sam poit (D o D) wh o voltag a applid. To simplify th xplaatio, oly fou ig pixls (o chals) a show, labld with,,, ad 3. Th light souc 6 has b wavlgth-discimiatd by a FPI. Wh oly voltags V ad V3 a applid to Chal ad Chal 3 of th fist SCPC, spctivly, Chal ad Chal a outd to th dstiatio D, ad Chal ad Chal 3 a tasmittd uimpdd dow th optical path. Fo th scod SCPC, if oly ig pixl has a applid voltag, Chal 3 is outd to dstiatio D, whil Chal passd though alog th optical axis. By applyig appopiat voltags to difft ig pixls of difft SCPC lmts, chals ca b adomly outd to ay dstiatio.

122 3 Fig. 5..Th coss-sctio dawig of a 4X switch mployig two idtical SCPC lmts. Fig. 5.3 shows a adom optical coss-switch ca b built by stacig multipl SCPC uits. Th collimatd light fom th Faby-Pot talo popagats ito th stac of idtical SCPC uits. Each SCPC uit covts a slctd wavlgth chal i th cicula itfc patt to a poit, which ca b pojctd oto a dtcto o outd to difft clit dstiatios though optical fibs. Fig A adom optical coss-switch by stacig multipl SCPC uits.

123 4 5.4 Optics Dsig of SCPC Two typs of SCPC hav b dsigd. Th fist typ of SCPC is a plai tasmissio HPDLC, whos fuctio is stig th bam. Th diffactd (std) bam is focusd by a focal ls to a poit, as dpictd i Fig I th scod typ of SCPC, th HPDLC, fuctioig as a ls, focuss th collimatd icidt light to a poit, as show i Fig. (5.). CH CH CH3 SCPC Focusig Ls Fig Th fist typ of SCPC: th diffactd bam is focusd by a focal ls to a poit. Lft: difft chals a outd away by th SCPC ad futh focusd by th focusig ls. Right: wh som chals (d ad blu i th figu) o th SCPC a switchd o, th switchd chals pass though th SCPC without big std to th dtcto Fist Typ (Bam Stig) SCPC I th fist typ of SCPC dvic, th HPDLC fuctios as a bam stig dvic with th sam diffactio agl θ d vywh i th HPDLC aa (Fig. 5.6). As th SCPC wos i th coditio of omal icidc, th HPDLC i th SCPC dvic quis that th Bagg coditio

124 5 (a) Diffactd bam (b) θ d Glass Λ θ t Λ Glass θ θ di θ w θ wi θ wi Icidt adig bam θ w θ w Rcodig bams Fig Radig bam cofiguatio (a) ad codig bam cofiguatio (b) of th bam stig HPDLC fo th fist typ of SCPC. b satisfid wh th icidt bam is omal to th cll sufac. Th adig bam (wavlgth λ ) ad diffactio bam cofiguatio is illustatd i Fig Th HPDLC gatig is codd by xposig th cll to two itfig las bams whos wavlgth is λ w, as show i Fig. 5.5 (b). Th codig bam icidt agls ca b dtmid, povidd that th factiv idx of th HPDLC mixtu at th codig wavlgth λ w, w, ad th avag factiv idx of th HPDLC at th adig wavlgth λ, is ow. Th diffactio agl isid th HPDLC θ di is dtmid by Fsl's law: = siθ acsi d θ di. (5-)

125 6 Sic th icidt bam is ppdicula to th cll sufac, th gatig pla tilt agl θ t is th sam as th agl btw th gatig pla ad th icidt adig bam θ, thfo, θ = θ = θ. Bagg's law quis that di t Λsiθ Λsiθ w = = λ, λw, w (5-) wh θ w is th agl btw th codig bam ad th gatig pla. Thfo, λ w θ = acsi si w θ. (5-3) λw Th codig bam icidt agls isid th HPDLCs θ ad θ i, which a dfid i Fig. (5.6b), a th giv by wi w θ θ wi wi = = θ + θ, t θ θ. t w w (5-4) Th codig bam icidt agls i ai a fially dtmid: θ θ w w = = acsi( w acsi( w siθ siθ wi wi ), ). (5-5) o If w st th diffactio agl of th HPDLC to b θ = 79, a lag agl i od to miimiz th distac btw ighboig SCPC dvics wh thy a stacd togth, ad assumig that o =.5, th codig bam icidt agls ca b divd: = w = θ w θ w o ad = 5.

126 Scod Typ (Focusig) SCPC I th scod typ of SCPC dvic, th HPDLC i th SCPC dvic is dsigd to focus th light to a poit, fuctioig li a hologaphic ls that was dmostatd by Ritch ad cowos i 974 [63]. W costuctd a itfc patt usig a poit souc ad a pla wav fo fabicatig th HPDLC with a built-i focus fo th scod typ SCPC. Th hologaphy stup is illustatd i Fig HPDLC cll Mio Iis Bam Splitt Shutt Bam Expad Focal Ls 53 m Las Fig Th hologaphy stup fo fabicatig th scod typ of SCPC. Th poit souc is gatd by addig a focal ls i th optical path of o codig bam. Th focal lgth of th ls is F, ad it is placd at a distac F fom th sampl cll. Th codig ad adig optics of th HPDLC acoss th ct of th HPDLC aa a simulatd. Th codig bam cofiguatio a th sampl is illustatd i Fig. 5.7.

127 8 θ w A D A B C θ F θ w B θ wc w A θ w B θ w C Fig Rcodig bam pofil acoss th HPDLC aa usig th stup show i Fig Astigmatism i Scod Typ (Focusig) SCPC Fo th SCPC woig at a wavlgth (such as 54 m) difft fom th codig wavlgth (such as 53 m), astigmatism a xplaid i ou followig simulatio ad calculatio, ad also cofimd i ou tstig sults. I ou modlig, w still st th diffactio agl at th ct of th HPDLC to b o θ = 79 ad w =.5; th codig bam icidt agls at th ct of th HPDLC a: = o o θ = ad = 5, as calculatd i th pvious sctio. Th diamt of th w θ w ffctiv HPDLC aa is D =.54 cm. Fo th collimatd codig bam, th icidt agl is o idtical at difft locatios of th HPDLC: θ = θ = θ = 5. Th icidt agls w A w B w C of th divgig codig bam at difft locatios a calculatd fo difft F. Th icidt agls at spot A ad spot C a pstd i Tabl 5., whth valus a dfid by:

128 9 θ : th Bagg matchd icidt agl fo 54 m at A; : ba agl fo 54 m at C; : dba diffactio agl at C; : da θ th Bagg matchd icidt θ th Bagg diffactio agl at poit A; : bc θ th Bagg θ th diffactio agl with 54 m omal icidc at A; θ : th diffactio agl with 54 m omal icidc at C; TIR: total ital flctio; d: miimum distac btw ighboig lctods dtmid by th diffactd bam cofiguatio. With th icidt agls of th codig bams dtmid, w calculat th Bagg matchd icidt agl ad diffactio agls fo th 54 m adig bam o difft locatios acoss th HPDLC aa. Alog th li ABC as show i Fig. 5.7, th omal icidt light at poit B, ct of th SCPC, is o doubt Bagg-matchd; howv, as th icidt locatio movs away fom th ct B, th omal icidc is o mo Bagg matchd. At poits A ad C which show th lagst dviatios fom omal icidc, th Bagg matchd icidt agls θ ba adθ bc, ad th cospodig diffactio agls θ dba ad θ dbc dbc dc, a calculatd ad listd i Tabl 5.. Cosidig th agula dpdc of th diffactio fficicy discussd i Chapt, ou calculatd sult idicats: () Th diffactio fficicy fo omal icidt of 54 m light dcass with th icas of th distac fom th icidt spot to th ct of th HPDLC. () Th dviatio of th Bagg matchd icidc fom th omal icidc dcass with icasig F.

129 F (mm) θ wa ( ) θ wc ( ) θ ba ( ) θ dba ( ) θ bc ( ) θ dbc ( ) θ da ( ) θ dc ( ) d (mm) TIR TIR N/A TIR TIR N/A TIR TIR N/A TIR TIR N/A TIR TIR N/A TIR TIR N/A TIR TIR N/A TIR TIR N/A TIR TIR N/A Tabl 5.. Covgig codig bam icidt agls, Bagg adig ad diffactio agls, diffactio agls with omal icidt adig, ad miimum distac btw ighboig SCPC uits.

130 Fo poits A ad C, th diffactio agls θ da adθ dc fo omal icidt adig at 54 m a also calculatd ad listd i Tabl 5.. Ths agls a ot Bagg matchd. It is impotat to otic that wh F is lss tha mm, th diffactio bam at th lft dg, A, will b totally flctd at th glass-ai itfac ad tappd i th HPDLC util fially scapig fom th dg. This total ital flctio (TIR) ffct futh dcass th diffactio fficicy at th dg of th HPDLC aa. To avoid th TIR ffct, th focal lgth of th ls should b lag tha mm. th Wh th diffactio agls a ow, th covgig poptis of th diffactd bam i x y pla ca b divd fom Bagg's law. Fig. 5.8 shows th covgig poptis of th diffactd bams acoss th ct of th cll i th y dictio. Th focus is ot idal i th x y pla. With th icas of F, th waist of th diffactd bam (th thist width of th diffactd bam) dcass, ad th positio of th bam waist movs away fom th HPDLC. To obtai a lativly good focus, a ls with a lag focal lgth is pfd i th codig stup. Sic th ls is placd at F fom th HPDLC duig xposu, a lag F bigs icovic to th fabicatio pocss. Two focal lss i sis ca b usd to gat th sam divgig codig bam pofil at th positio of th HPDLC. I th simulatio abov, oly th locatios acoss th ct of th HPDLC aa a cosidd. A qualitativ simulatio of vaious locatios ov th ti HPDLC aa usig ZEMAX, a optical simulatio ad dsig softwa, vals that th focusig of th diffactd bam is also astigmatic, which matchs wll with ou calculatio ad aalysis. Th covgc of th diffactd bam i th x y pla is fast tha that i th z dictio. Thfo th lia dimsio of th focus poit is lag tha that of th bam waist calculatd i th simulatio abov.

131 y F = A B C x y F = A B C x y F = 3 A B C x Fig Th diffactio bam pofil of ich HPDLCs fabicatd usig lss with vaious focal lgth F.

132 3 5.5 Fabicatio ad Chaactizatio of SCPC Woig i Visibl Wavlgths 5.5. Sigl Chal SCPC Fo SCPCs woig i visibl wavlgths, w bga sampl fabicatio with th opixlatd HPDLC fo th scod typ of SCPC dvics, o sigl chal SCPC dvics with built-i covgc, usig ITO glass substats. A UV-cuig mixtu cosistig of 5% PN393 ad 5% liquid cystal TL 5 was ppad. Th HPDLC mixtu was sadwichd btw two substats with th cll gap cotolld by 5 μ m fib spacs. Th cll was xposd to a 35 m A + io UV las fo 9 scods i hologaphic stup as illustatd i Fig Th itsity of a sigl codig bam was ~ 6 mw/cm. Zo Od Focal Poit Lft: N o voltag applid; Right: V oltag applid. ( V A C squa-wav HZ) Fig Th lft pal : th switch-off stat of th SCPC (o voltag applid); th ight pal : th switch-o stat (voltag applid). I ach pal, th hologaphic focal poit is th poit o th ight sid, ad th pass-though light is o th lft.

133 4 Fig. 5.9 shows digital imags of a HPDLC sampl woig with a d las (63.8 m), showig th switchig of th focal poit of th SCPC. Th imag o th lft pal is th switchoff stat of th SCPC wh o voltag is applid, ad th imag o th ight pal is th switcho stat wh a V squa-wav AC KHz sigal is applid. O ach pal, th lag light spot is th pass-though light, o zo od, ad th lft fi poit is th focal diffactd light. Th diffactio agl away fom th tasmissio bam is 45. Th hologaphic focal poit was sampld with a dtcto, ad th cotast atio of th switch-off vsus switch-o stat is 5:. Th ovall fficicy of th hologam was 3%. Th loss of th fficicy coms fom th scattig of th matials, ad flctio fom th sufacs as o AR coatig is usd o th glass substats. W also dvlopd a fomula fo g-las (53 m) cuig. Th matials a vaious atios of Ebcyl 83 ad 4866 as p-polym bass to cotol th ffctiv fuctioality of th polym. Ths a mixd with th matic liquid cystal BL38 ( =.57, =.799) fom EM Idustis. Ros Bgal ad N-Phylglyci w slctd as th photoiitiato ad coiitiato that a ssitiv to th visibl light. Th solutio of photoiitiato was ppad with 4. wt.% Ros Bgal ad. wt.% N-Phylglyci i N-Viyl Pyolidio. Th sufactat Sobita Moo-Olat was also addd to th mixtu, which is ow to duc th sufac itactio stgth btw th liquid cystal ad polym. Th wight atio of Oligom: liquid cystal: iitiato solutio: sufactat was 45: 3.4:.6:. A doplt of th p-polym mixtu was sadwichd btw two AR-coatd ITO glass substats with 5-mico spacig thicss cotolld with glass fib spacs. Th SCPC sampl is xposd to a 5W g las at a wavlgth of 53 m fo 3 scods. Fig. 5. shows two imags of a sigl chal SCPC dvic i opatio fabicatd with g-las cuig. Th bam siz of th adig las (53 m) is about o-half ich, show as o

134 5 th lag spots i both lft pal ad ight pal. Th SCPC covts th icidt light ito a fi poit (th small spots i both imags) that is 45 off axis fom th ct of th SCPC. Th focusig ffct of th SCPC is vy good, with a focus lgth of cm, matchig th witig coditio. Th ight imag i Fig. 5. shows th SCPC switchd by a AC fild of 6 Volts, Hz. A Mlls Giot las pow mt was utilizd to masu th itsity of th icidt bam (I i ) ad th tasmittd bam (I th ), ad a OPHIR ifad pow mt was mployd to masu th diffactd bam itsity (I st ). Th tasmittac was calculatd as I th / I i, ad th diffactio fficicy was calculatd as I st / I i. Th tstig sult a dtaild i Tabl 5.. Th cotast atio of switch-off to switch-o is lag tha 4. Tasmittac I th / I i Diffactio Efficicy I st / I i Switch O 95% 5% Switch Off 5% 8% Tabl 5. Tstig sult of SCPC. Fig. 5.. Switchig of a SCPC woig at 53 m.

135 chal SCPC Th SCPC is dsigd to wo i tadm with a Faby-Pot talo to covt th cicula itfc fig patt ito poits o a poit aay. Thus th ig pixls of th ITO patt o o SCPC substat must match th Faby-Pot ig. Fig. 5. shows a ITO ig patt fo a -chal SCPC. Fig.5.. A schmatic dsciptio of CAD dsig of a -pixl ITO patt i SCPC Th fabicatio of a multipl chal SCPC is simila to th fabicatio of o-pixl SCPC, bcaus th hologaphic patt i all th pixls a witt at th sam tim. A -pixl SCPC woig fo 53 m was fabicatd. Fig. 5. shows th switchig of th ct pixl of th - pixl SCPC that focuss chals to o poit. O both th lft ad ight imags, th lft light spot (bigg spot) is th light that pass though th SCPC, th ight spot (small) is th focusig poit. I th lft imag, th ct pixl is switchd o with a voltag of 5 Volts. I th ight imag, o voltag is applid. Bcaus all of th pixls focus o o poit, th is still light o th focusig poit wh th ct pixl is switchd o.

136 7 Fig. 5.. Switchig of th ct pixl of -pixl typ-ii SCPC. Fig Switchig of o o-ct pixl of -pixl typ-i SCPC W also dmostat th -pixl switchig of th bam-stig SCPC (typ I) that dicts all chals to difft dictios without focusig. A additioal focusig ls (ot show i th figu) aft th SCPC is dd to focus th slctd chals to o poit. Fig. 5.3 shows th switchig of o ig-pixl of th bam-stig SCPC. Th lft poit shows th light that is

137 8 std by th SCPC, th ight poit is th light passig though. Th lft poit ad th ight poit hav th sam siz. Wh th ig-pixl is switchd o with a voltag of 5 volts, all of th sigal fom th slctd ig-pixl pass though without big std to th lft poit, so a blac ig appas i a bight bacgoud that is th oth 9 chals (icludig th ct pixl). Fig. 5.4 shows th switchig of th ct pixl of a bam-stig SCPC. Fig.5.4 Switch o th ct pixl of a bam-stig -chal SCPC. 5.6 Fabicatio ad Chaactizatio of SCPC Woig i NIR Wavlgths 5.6. Matial Optimizatio fo Big-Aa SCPC Woig i NIR Th HPDLC matials ca b dividd ito two catgois: UV cuabl matials ad visibl cuabl matials. Both matials hav b usd to dmostat th SCPC cocpt woig i th visibl wavlgth ag. Wh fabicatig th SCPC with a lag aa (miimum.54 cm diamt) woig i th 55 m wavlgth ag, a 53 m las istad of a UV las is usd

138 9 fo th hologaphic codig, du to th lac of sufficit pow i th UV las, ad th high fficicy of visibl cuabl matials. Matial optimizatio was iitiatd fom sval bas fomulas with difft mooms ad sufactats. Th liquid cystal i all ths fomulas is BL38, a liquid cystal mixtu fom EMD Chmicals that has a high optical bifigc (.7). Th cotts of th HPDLC mixtus a listd i Tabl 5.3. Th moom mixtu i Fomula cosists of th utha acylat mooms Ebcyl 83 ad Ebcyl 4866, both of which a poducd by UCB Chmicals. Sobita moo-olat, S-7 by Chm Svics, is utilizd as a sufactat to duc th switchig voltag. Th moom mixtu i Fomula cosists of Ebcyl 83 ad Timthylolpopa tis (3-mcaptopopioat) (TT3) fom Sigma-Aldich. A fluoiatd acylat moom,,,,3,3,3 Hxafluooisopopyl Acylat (HFIPA) is utilizd as a sufactat i Fomula. Th moom i Fomula 3 is Diptaythitol Ptaacylat (DPHA), a tta-fuctioal moom. A moo-fuctioal moom, Viyl Nooaoat (VN), svs as th sufactat i this fomula. Photoiitiato solutios a quid i all of ths fomulas to abl th photo polymizatio ad fo solvig th liquid cystal i th mooms. Th photo iitiato solutios cosist of Ros Bgal (photoiitiato), N-Phylglyci (NPG, co-iitiato) ad -Viyl--Pyolidio (NVP, solvt ad chai tmiato), all availabl fom Sigma- Aldich. Th HPDLCs fabicatd usig ths matials w chaactizd ad sval issus aos. As th opatio wavlgth of th HPDLC is ~5 m, a 5-mico cll gap is cssay to achiv sufficit diffactio fficicy. Th switchig voltag cssay to compltly dactivat th HPDLC is high (> 5V), ad th sampls may xpic a dilctic badow wh a high voltag is applid acoss th cll. Th sufactat i Fomula itoducs ios ito th HPDLC ad substatially icass th coductac of th HPDLC. Th HPDLCs fabicatd usig Fomula matial a asy to hat up wh a voltag is applid ad xpic thmal

139 switchig istad of lctical switchig. Sigificat ffot was mad to sach fo a appopiat HPDLC fomula that solvd ths issus. Wh th liquid cystal BL is usd to plac th liquid cystal BL38 i all th fomulas, a ductio i th switchig voltag was obsvd; howv, th diffactio fficicy also dcasd substatially. Sigs of thmal switchig w also obsvd i som of th sampls. It is cla th BL fomulas w ot suitabl fo th dvic. Matials basd o th th oigial fomulas but with vaid compot atios w futh ivstigatd. Fomula Fomula Fomula 3 Ebcyl 83 Ebcyl 4866 BL38 P.I.Solutio S-7.5%.5% 3.4%.6%.% Ebcyl 83 TT (3) BL38 P.I.Solutio HFIPA 36% 4% 34% 6% % DPHA BL 38 P.I.Solutio VN 47% 38% % 5% Ros Bgal NPG NVP P.I.Solutio 4% % 86% P.I.Solutio 3% 7% 9% Tabl 5.3. Compots of th HPDLC mixtus iitially ivstigatd. Aft a sis of xpimts, a HPDLC mixtu that is suitabl fo th HPDLC opatig at ~ 5m basd o Fomula 3 is dvlopd. Th cotts of th w fomula (Fomula-SCPC) a pstd i Tabl 5.4.

140 Fomula-SCPC DPHA BL38 P.I.Solutio VN 35.9% 43.36%.45%.7% Tabl 5.4. Matial cotts of Fomula-SCPC Fabicatio ad Chaactizatio of Sigl Chal SCPC Woig i NIR As w hav discussd i th optical dsig of th SCPC, th typ-ii (focusig) SCPC woig at 55 m has stog astigmatism, which is also cofimd i tstig of ou typ-ii sampls woig at 55 m. Th focusig i th x y pla was fast tha i th z dictio, which is i agmt with th sults of ou qualitativ simulatio. Ou sach focuss o typ- I (bam-stig) fo SCPC dvics woig at 55 m. Bam Splitt Sampl Mio Iis Shutt Bam Expad θ w θ w 53 m Las Fig Hologaphic codig stup fo fabicatig th SCPCs woig i th 55 m ag. W fabicatd th sigl chal HPDLC usig ITO-coatd glass substats with AR coatig. Th Fomula-SCPC HPDLC mixtu was sadwichd btw two substats. Th cll gap was cotolld by 5 μ m fib spacs. Th cll was xposd i th hologaphic stup illustatd i Fig. 5.5 fo 9 scods. Th codig bam icidt agls at th HPDLC w:

141 o o o θ = ad = 5 to abl a lag diffactio agl of θ = 79. Th itsity of a w θ w sigl codig bam was 6 mw/cm. W chaactizd th lcto-optical poptis of th HPDLCs fabicatd fo th SCPC dvic. All of th HPLDCs showd stog polaizatio dpdc. Th diffactio fficicy fo s-polaizd light was much lss tha that of th p-polaizd light ad was thfo glctd, oly p-polaizd light was tstd. A KHz squa wav sigal was usd to addss th HPDLCs. Th tasmittac ad diffactio fficicy as a fuctio of voltag a pstd i Fig Th switchig voltag was substatially dcasd ad th diffactio fficicy was 55%. Elcto-Optical Rspos of SCPC.6.5 Efficicy Tasmissio Diffactio Voltag ( v ) Fig. 5.6.Tasmittac ad diffactio fficicy as a fuctio of voltag. Compad with th high diffactio fficicy (8%) of th SCPC woig at 53 m, a umb of factos low th diffactio fficicy of th SCPC opatig at 55 m: () Th Pitch of gatig of th HPDLC is much lag; () Th AR coatig is optimizd fo 53 m, ot 55 m;

142 3 (3) Th Shap diffactio agl futh dgads th AR coatig pfomac; Fabicatio ad Chaactizatio of 3-chal SCPC Woig i NIR Fabicatio Pocss Th 3-chal SCPC uits w fabicatd with AR ITO-coatd substats with thicss of.5 mm, thi tha th pviously usd glass substats with a thicss of. mm. Th pupos of usig th thi glass is to miimiz th siz of th total dvic wh multipl SCPC uits must b stacd togth. To hold th clls of th thi substats poply duig xposu, a spcial sampl hold was dsigd ad fabicatd. Th hologaphic codig stup fo fabicatig th SCPC uits is illustatd i Fig Th two codig bam icidt agls w st to b θ w =4 5, ad θ w icidc adig bam to b diffactd to a lag agl of o θ = 79 =8 5, fo th omal. Th Fomula-SCPC matial was utilizd. 5-mico fib spacs cotolld th cll gap. Th clls w pssd at 4.5 psi by a balloo fo miuts to su cll gap homogity pio to th hologaphic xposu. Th xposu tim was 5 miuts ad th total output pow of th las was st to 3W duig th xposu. Aft xposu, th cll was cud i a 3W las bam to polymiz th mooms outsid th HPDLC aa. Th dgs of th clls w scud with 5-miut poxy Switchig of th 3-chal SCPC Th switchig poptis of th SCPC w ivstigatd. Th divig sigal was a Hz squa wav gatd by a HP fuctio gato ad a T amplifi. Voltag was applid o ach of th 3-chals of th SCPC uit o chal at a tim, ad th switchig of ach sigl ig pixl was obsvd. Fig. 5.7 dmostats th switchig of sval chals of a SCPC sampl.

143 4 Fig Switchig of idpdt chals i th SCPC uit. Th photos, fom lft to ight, show th dactivatio of th ctal pixl, th 5 th pixl (cout fom th ct), ad th outmost pixl (3 th ), spctivly. Th lcto-optical pfomac of th ctal pixl of this SCPC uit is show i Fig Th tasmittac of som chals is summaizd i Tabl 5.5. Th diffactio fficicy was low tha that of th HPDLC fabicatd usig upattd glass substats. This is attibutd to th scattig fom th tchd lis ud las xposu ad th ouifomity causd by th thi glass, which futh dgad th itfc patt of th codig bams ad low th gatig quality.

144 5 Elcto-optical Rspos of SCPC uit (%) Applid Voltag ( v ) Tasmissio Diffactio Fig Th omalizd tasmittac ad diffactio fficicy of th ct chal of a SCPC uit as a fuctio of voltag. Chal Numb To (%) Toff (%) Tabl 5.5. Th tasmittac of som chals of a SCPC uit.

145 Wavlgth Dpdc Th SCPC dvic was dsigd to opat i th wavlgth ag of m. Thfo, th wavlgth dpdc of th dvic pfomac was a citical facto. A Agilt 8689A tuabl las was utilizd to chaactiz th wavlgth dpdc of th 3- chal SCPC dvics. Th masumt stup is illustatd i Fig Th icidt light was ppdicula to th SCPC cll sufac. Th itsity of th icidt light, th tasmittd light, ad th diffactd light was masud usig a Mlls Giot uivsal pow mt at vaious wavlgths. Th zo fild tasmittac ad diffactio fficicy of th SCPC dvic w calculatd ad th sults a pstd i Fig. 5.. Photo Dtcto Collimatio Ls Iis I st Agilt 8689A Tuabl Las I i I th Fig Optical stup fo masuig th wavlgth dpdc of th SCPC uits.

146 7 Fig. 5.. Tasmittac ad diffactio fficicy as a fuctio of icidt wavlgth of th switch-off stat of a SCPC sampl: JL44B. Th SCPC sampls hav stog polaizatio dpdc; th diffactio fficicy of s- polaizd light is substatially small tha that of p-polaizd light ad thfo ca b glctd. Th icidt light i th masumt was p-polaizd. Th diffactio fficicy ad tasmittac w calculatd i fc to th icidt light itsity. Th wavlgth of th icidt light was icasd fom 55 m to 575 m, i icmt stps of 5 m. Th tasmittac of th SCPC shows o substatial chag with th wavlgth; howv, th diffactio fficicy dcasd substatially wh th icidt wavlgth was gat tha 56 m. Th dcas is attibutd to th icas of th diffactio agl with a icas i th adig bam wavlgth. Wh th adig bam wavlgth is icasd fom 58 m to 56 m, th diffactio agl chags fom to 83 37, which is vy clos to total ital flctio. Th pfomac of th ati-flctio coatig dgads substatially wh th icidt agl is clos to TIR. Wh th icidt wavlgth is 575 m, th icidt light is

147 8 totally flctd at th glass-ai itfac. Th diffactio fficicy is fittd with th coupld wav thoy fo a tasmissio gatig, η = A A ( L) () cosθ cosθ = κ Δα κ + Δα si κl + κ /. (5-6) wh = π cos(θ ) / λ cos( θ ), B B π Δλ Δα = Λ Λcosθ B is th phas mismatch gatd fom th dviatio i wavlgth. As th codig bam icidt agls a: θ w =43 59 ad θ o = 5 w as show i Fig. 5.6(b), th witig agl θw isid th HPDLC is dtmid by: θ = ac(si( θ w )/.5) - acsi(si( θ w )/.5); θ w = Cosidig a witig wavlgth w of λ=53 m ad factiv idx ~.5, th piod of th gatig Λ is Λ siθ w = λ ad calculatd to b Λ =.478 μm. Fo a adig bam with omal icidc o th cll sufac, as th dsigd xit agl i ai is 79, th Bagg agl is θ B =acsi((si(79 ))/.5), o θ B =.44.(s Fig. 5.(a)). Th wavlgth that mts th Bagg coditio is: λb = Λsiθ B =.548 μm. As th wavlgth vais fom λ B, th diffactd agl shifts to mt th coditio of costuctiv itfc: Λ siθ + Λ siθ = λ/, as th gatig ca b tatd as a thi gatig (dfid i Chapt ). Wh th diffactd light passs though th glass-ai bouday, th itsity of th factd light is [64]: I = si θ si θ (5-7) di d // ( ) si ( θdi + θ d)cos ( θd θdi ) Cosidig th coupld wav thoy ad th light factio quatio (5-7), w fit th modlig sult with th masumt sult, as show i Fig. 5.. Th modl dos ot cosid

148 9 th ffct of scattig, which may xplai th discpacy btw th modl ad th xpimtal sult. Diffactio Efficicy a Wavlgth Dpdc of Diffactio Efficicy.5..5 Modlig Masumt Wavlgth Fig. 5.. Fit of th modl basd o coupld wav thoy ad th factio picipl, with th xpimtal sult, fo th wavlgth dpdc of th diffactio fficicy Agula Dpdc Th dpdc of th tasmittac ad diffactio fficicy o th adig bam icidt agl was chaactizd to valuat th hologaphic codig optical stup. A 54 m fib las gatd th icidt bam. Th tasmittd bam itsity ad diffactd bam itsity w masud usig a Mlls Giot uivsal pow mt ad a OPHIR ifad pow mt with vaious adig bam icidt agls. Th tasmissio as a fuctio of icidt agl is pstd i Fig. 5., ad th diffactio fficicy as a fuctio of icidt agl of th SCPC is pstd i Fig Both Fig.s 5. ad 5.3 a basd o th tst sult of SCPC sampl

149 3 JL44B. Th diffactio fficicy achd its maximum at omal icidc, whil th tasmittac miimizd at a icidt agl of. This sult povd th choic ad cotol of codig bam icidt agl was optimizd fo omal icidc adig of th dsigatd wavlgth. Wh th adig bam icidt agl was gat tha, th diffactio fficicy dcasd much fast tha th icas of th tasmittac. Th abuptss of th dcas i diffactio fficicy was du to th icas of th diffactio agl with th icas of th icidt agl. Th dsigd diffactio agl i th dvic was 79 i ai ad th diffactio agl isid th SCPC was clos to th total flctio agl at th glass-ai itfac. Th fficicy of th ati-flctio coatig o th glass substats dcasd substatially wh th diffactio agl was clos to th total ital flctio agl ad th diffactd light would b totally flctd wh th total ital flctio agl was achd o xcdd. Tasmissio Vs Icidt Agl Tasmissio Icidt Agl Fig. 5.. Th tasmissio as a fuctio of icidt agl of th SCPC.

150 3 O th oth had, th pfomac of th ati-flctio coatig hadly chagd fo th tasmittd light. Th aso that th tasmittac chags with th icidc agl ca b mostly attibutd to th chag of diffactio fficicy with th chag of icidt agl. Cosidig coupld wav thoy, th diffactio fficicy η fo tasmissio gatigs ca b fit usig th tasmissio data: / κ Δα η = si κl + Δα κ ( ) κ, + wh = π cos(θ ) / λ cos( θ ) is th couplig costat, ad Δ α = π Δθ siθ / λ B B 4 B is th phas mismatch. H λ is th adig bam wavlgth, θ B =.44 is th Bagg agl, Diffactio Vs Icidt Agl Diffactio Icidt Agl Fig. 5.3.Th diffactio fficicy as a fuctio of icidt agl of th SCPC.

151 3 =.6 is th avag factiv idx of th HPDLC, is th idx modulatio of th gatig, ad Δ θ is th dviatio fom Bagg agl. Th cospodig tasmittac is giv by -η, povidd that all oth losss a glctd. Th masumt was fo p-polaizd light oly; thfo, th couplig costat κ was calculatd fo p-polaizd light ad th fittig was oly valid fo p-polaizd light. Th bst-fit cuv is pstd i Fig Th idx modulatio of th HPDLC is =. ±.. Tasmissio Vs Icidt Agl Nomalizd Tasmissio Icidt Agl Modlig sult Masumt Fig Nomalizd tasmittac a fit to th fomula fo a tasmissio gatig divd by coupld wav thoy. Masumt sults a of SCPC sampl JL44B.

152 Summay ad Coclusios W hav dmostatd th cocpt ad dsig of a iovativ SCPC dvic, ad hav fabicatd ad tstd SCPC dvics woig at difft wavlgths (visibl ad NIR), ad with difft chal umbs (sigl chal, chal, ad 3 chal). Two typs of SCPC dvics a aalyzd with mo focus o th scod typ, a bam-stig SCPC. Th high diffactio fficicy of up to 8% i th visibl, ad 6% i th NIR was achivd. Th wavlgth ad agula dpdc w also ivstigatd. This sach illustats th pottial fo maig lctically tuabl optical dvics such as adom optical switchs ad spctal imagig dtctos.

153 CHAPTER 6 Lasig of Dy-Dopd HPDLC 6. Itoductio I this chapt, w will discuss th matials, fabicatio ad chaactizatio of lasig missio i dy dopd HPDLCs. Lasig fom difft mods of HPDLCs will b studid ad both th switchig ad tuability of th lasig fuctio will b dmostatd. Lasig fom twodimsioal HPDLC basd photoic bad gap (PBG) matials will also b dmostatd. Fially, lasig fom polaizatio modulatd gatigs will b discussd. 6. Itoductio to Dy Typically, dy molculs a mo o lss od-li. Usually th majo compot of th tasitio momt of th molcul is alog th log molcula axis (positiv dy) o shot axis (gativ dy) [65]. Positiv dys, as i Fig. 6.(a), absob th compot of upolaizd light i th log axis of th molculs. I tms of absobac A, A > A. I gativ dys (s Fig. 6.(b)), A < A ad absoptio occus othogoal to th molcula axis. Sic th molcul is // otatig, light is absobd i ay dictio othogoal to th axis. Th dichoic atio D ad is dfid by: // D = A A // (6-) Claly, fo a positiv dy, D > ad fo a gativ dy, D<. Th most widly usd dichoic dys i gust-host LCDs fall basically ito two classs fom a chmical stuctu poit of viw azo dys ad athaquio dys. Fig. 6. dscibs th 34

154 35 molcula stuctu of two ids of dys. I Fig. 6.(a), A psts a accpto, such as NO3 ad D psts a doo such as NH. Without A ad D, th dy absobs light i th UV bad, whil with difft accptos ad difft doos, th azo dys will absob light i a difft wavlgth bad. May azo dys hav b foud to absob light i th visibl wavlgth bad. Fig. 6.(b) shows a basic athaquio dy. Without ay substitut, th dy absobs UV light. To ma a dy that absobs visibl light, substituts a itoducd at,4,5 o 8. (a) (b) Fig..6.. Absoptio of positiv dy (a) ad gativ dy (b). Usually azo dys hav high solubility i liquid cystals tha athaquio dys bcaus azo dy molculs a mo od-li tha athaquio dy molculs. To icas th solubility of athaquio dys i liquid cystals, usually doos such as NH ad accptos li NO o CN a o 7. A dichoic mixtu is basically a homogous mixtu of dy(s) i a liquid cystal host. Th vaious physical poptis of dichoic mixtus dpd upo th physical poptis of th dys, th host, ad th combiatio. Fo xampl th colo of th mixtu is maily dpdt o th

155 36 dys, whil th dilctic aisotopy, lastic costats ad factiv idics a basically thos of th liquid cystal. Viscosity is dpdt o both. Som of th poptis such as absobac ad pct tasmittacs a also dpdt o aligmt, cll gap, tc. Th additio of a dy may slightly modify th physical poptis of th liquid cystal mixtu such as its opabl tmpatu ag, dilctic ad optical aisotopy, tc. 3 4 A N N D O O (a) (b) Fig. 6. Two basic ids of dys (a) azo dy (b) athaquio dy Th dicto of dys i th host liquid cystal coicids with th dicto of th host, ). Howv, th dictio of ach dy molcul dviats fom th dicto ) du to thmal fluctuatios. Th impact of th thmal fluctuatio may b difft i th dy ad liquid cystal molculs dpdig o thi molcula lgths ad gomty, which a show i Fig Th liquid cystal (Lm) ad dy (Dm) molculs mas a agl θ ad φ spctivly with th dicto ). Th od paamts of th liquid cystal molcul ( S ) ad dy molcul ( S ) ad th tasitio momt of th dy absoptio ( log molcula axs a giv by: S T L ) as dtmid fom th distibutio of thi D

156 37 3cos θ S L = (6-) 3cos φ S D = (6-3) S T = 3cos θ T (6-4) Wh θ ad φ a th agls mad by th log molcula axs of th liquid cystal ad dy molculs, spctivly, with th dicto of th liquid cystal (). θ T is th agl btw th tasitio momt ad th dicto. If w assum that th dictio of th tasitio momt, T, of th dy dviats fom its log molcula axis, Dm, at a agl β: th absobac, A, of th icidt polaizd light whos lctic fild vcto, E, mas a agl, ψ, with th dicto,, ca b giv by [65]: ( 3si β ) cos sd SD SD A ( β, ψ ) = cd si β + + ψ (6-5) 3 Wh is th magitud of th tasitio momt ad c ad d a, spctivly, th coctatio of th dy ad th thicss of th liquid cystal lay. Th dichoic atio is xpssd as th atio of th absobac at ψ= ad ψ=π/: D = A A // = A( β; ψ = ) + 4S = π S A( β; ψ = ) D D 6S + 3S D D si si β β (6-6) It is itstig to ot that (6-5) ca b witt as A( β, ψ ) = A + ( A// A )cos ( ψ ) (6-7) Th od paamt, S T, of th tasitio momt is dtmid xpimtally as:

157 38 S T = A A // // A + A = D D + (6-8) Fom (6-6) ad (6-8) w gt: S ( 3si β ) S = D T (6-9) Fo logatd plochoic dys, β is xtmly small, so S = S. Th lat discussio will T D us S to pst both th dy od paamt ad th od paamt of th tasitio momt wh w assum β=. Dicto φ Dm: Axis of Dy Molcul β T: Tasitio Momt θ Lm: Log Axis of liquid cystal Molcul ψ P: Elctic Vcto of Polaizd Light Fig Dy molculs isid liquid cystals

158 Itoductio to Las Th wod "las" is a acoym fo Light Amplificatio by Stimulatd Emissio of Radiatio. A taditioal las is composd of a pumpig souc, a activ las, o gai, mdium ad a soat optical cavity. Th gai mdium svs to tasf th xtal gy fom th pump souc ito th las bam. Th simplst las modl to udstad is th two gy-lvl systm, with a goud stat gy ad a xcitd stat of gy, wh >, as show i Fig. 6.4(a). Assum th is a goup of N atoms; th umb of ths atoms i th goud stat is dfid as N, whil th umb i th xcitd stat is N, satisfyig N + N = N. Th gy diffc btw th two stats is giv by ΔE = E E = h ν, Wh h is Plac's costat, ad ν dtmis th chaactistic fqucy of light that itacts with th atoms. I a thmal quilibium stat, th atio of th umb of atoms i ach stat is giv by a Boltzma distibutio[3, 3]: N N ( E E) = xp[ ], (6-) T wh T is th tmpatu (i uit of lvi) of th goup of atoms, ad is Boltzma's costat. Th a th possibl itactios btw a systm of atoms ad a light that w must cosid: Spotaous missio, absoptio, ad stimulatd missio. If a atom is i a xcitd stat, it may spotaously dcay to th goud stat at a at popotioal to N, thby mittig a photo of fqucy ν. H, th photos a mittd stochastically ad th is o fixd phas latioship btw th photos mittd fom a goup of xcitd atoms; i oth wods, this spotaous missio is icoht. I th absc of oth pocsss, th umb of atoms i th xcitd stat at tim t, is giv by:

159 4 t N ( t) = N () xp( ), (6-) τ wh N () is th umb of xcitd atoms at tim t=, ad τ is th liftim of th tasitio btw th two stats. This missio is dfid as a spotaous missio. E 3 E R (adiatiolss tasitio) E E hν P (pump tasitio) L (las tasitio) E (a) two-lvl systm (b) th-lvl systm Fig (a) Two-lvl gy systm of las mdium. (b) th-lvl gy systm. If light (photos) of fqucy ν pass though th goup of atoms, th xists a dfid pobability of atoms i th goud stat absobig a photo ad big xcitd to th high gy stat. Wh a atom i th xcitd stat itacts with a photo of fqucy ν, th atom may dcay, mittig aoth photo with th sam phas ad fqucy as th icidt photo. This pocss is ow as stimulatd missio. Citically, stimulatd missio is dfid by th fact that th iducd photo has th sam fqucy, phas, ad polaizatio as th iducig photo. I oth wods, th two photos a coht. It is this popty that allows fo optical amplificatio, ad th poductio of a las systm.

160 4 I th opatio of a las, all th light-matt itactios (spotaous missio, absoptio, ad stimulatd missio) occu. Iitially, atoms a gizd fom th goud stat to th xcitd stat by a pocss fd to as pumpig. If th goud stat has a high populatio dsity tha th xcitd stat (N > N ), th pocss of absoptio is domiat ad th is a t attuatio of photos. If th populatios of th two stats a th sam (N = N ), th at of absoptio of light xactly balacs th at of missio; th mdium is optically taspat. If th high gy stat has a gat populatio dsity tha th low gy stat (N < N ), th th missio pocsss domiat, ad th adiatio fild withi th systm udgos a t icas i itsity. I od to poduc a fast at of stimulatd missio tha absoptio, a populatio ivsio is quid: N /N >. I a two-lvl systm, th low gy stat cotais a lag populatio tha th high gy stat, as dscibd by quatio (6-), a populatio ivsio (N /N > ) ca v xist fo a systm i thmal quilibium. To achiv th cssay populatio ivsio, th systm must b pushd ito a o-quilibatd stat. At miimum, a th-lvl systm, as show i Fig. 6.4(b), is quid. Cosid a goup of N atoms with th gy stats, E, E ad E 3, ad E < E < E 3. Th populatio dsitis of ach stat a N, N ad N 3, spctivly. Iitially, th systm of atoms is at thmal quilibium ad th majoity of th atoms will b i th goud stat: i.. N N, N N 3. Wh th atoms a subjctd to light of a fqucy ν 3, wh E 3 - E = hν 3, th pocss of optical absoptio will xcit th atoms fom th goud stat to lvl 3, such that N 3 >. Th gy tasitio E E 3 is fd to as th pump tasitio. I a optical mdium suitabl fo las opatio, it is quid that ths xcitd atoms quicly dcay to lvl. Th gy lasd i this tasitio may b mittd as a photo

161 4 (spotaous missio), o, i pactic, th 3 tasitio (labld R i Fig. 6.4(b)) is usually adiatiolss, with th gy big tasfd to a vibatioal motio (hat) of th host matial suoudig th atoms. A atom i lvl may dcay by spotaous missio to th goud stat, lasig a photo of fqucy ν (giv by E - E = hν ), which is show as a las tasitio i Fig. 6.4(b). If th liftim of this tasitio, τ is much log tha th liftim of th adiatiolss 3 tasitio τ 3 (if τ >> τ 3 ), th populatio of E 3 will sstially b zo (N 3 ) ad a populatio of xcitd stat atoms will accumulat i lvl. If ov half th N atoms ca b accumulatd i this stat, th th populatio ivsio coditio (N > N ) is mt, ad optical amplificatio at th fqucy ν ca b obtaid. Though th fist typ of las to b discovd (basd o a uby las mdium, by Thodo Maima i 96) was a th-lvl systm, i pactic, most lass a fou-lvl systms, as dpictd i Fig H, th pumpig tasitio P xcits th atoms i th goud stat (lvl ) ito th pump bad (lvl 4). Th atoms i th upp lvl, E 4, ad low las lvl, E, dcay though fast, o-adiativ tasitios ito E 3 ad E, spctivly, ladig to gligibl populatio dsitis i th stats E 4 ad E : N ad N 4. Th las tasitio occus i th gy tasf fom E 3 to E. Sic th liftim of th las tasitio L is log compad to that of tasitios R ad R (τ 3 >> τ 43 ad τ 3 >> τ ), ay appciabl populatio accumulatig i lvl 3 will fom a populatio ivsio with spct to lvl. Thus, th optical amplificatio ad las opatio occus at a fqucy of ν 3 (E 3 - E = hν 3 ). As th light gatd by stimulatd missio is quivalt to th iput sigal i tms of wavlgth, phas, ad polaizatio, this givs las light its chaactistic cohc ad allows

162 43 it to maitai th uifom polaizatio ad moochomaticity stablishd by th optical cavity dsig, as was discussd i th chapt o Faby-Pot. E 4 R (adiatiolss tasitio) E 3 P (pump tasitio) L (las tasitio) E R (adiatiolss tasitio) E Fig A fou-lvl las gy diagam. Th lasig thshold is th lowst xcitatio lvl at which th las's output is domiatd by stimulatd missio ath tha by spotaous missio. Blow th thshold, th las's output pow iss slowly with icasig xcitatio. Abov this thshold, th slop of pow vs. xcitatio is ods of magitud gat. Th liwidth of th las's missio also bcoms ods of magitud small abov th thshold. It is abov th thshold that th las is cosidd to b lasig. Th lasig thshold is achd wh th optical gai of th las mdium is xactly balacd by th sum of all of th losss xpicd by th adiatio fild i o oud tip though th las's optical cavity. This ca b xpssd, assumig a stady-stat opatio, as [33]:

163 44 R R xp(gthl) xp( α l) (6-) = wh ad R a th mio flctivitis, l is th lgth of th gai mdium, g is th R gai paamt ad xp( g th ad xp( αl) is th oud tip pow loss. l) is th oud tip thshold pow gai, whil α is loss paamt Th optical loss is a a costat fo ay paticula las (α = α ), spcially clos to thshold. Ud this assumptio th thshold coditio ca b aagd as: th g th = α l( R R ) (6-3) l 6.4 Itoductio to Dy Las Ogaic dys a widly ow fo thi ability to gat las missio ov a wid wavlgth ag bcaus of th spcial lctoic gy lvls of th dy molculs [66]. I a dy molcul, ach lctoic lvl of th molcul is associatd with a st of vibatioal ad otatioal gy lvls spacd closly togth compad with th lctoic lvl spacig. Optical pumpig by xtal adiatio bigs th molcul fom o of th vibatioal-otatioal lvls of th goud lctoic stats to o of th vibatioal-otatioal lvls of a xcitd stat. Th xcitd dy molcul tds to dcay vy quicly to th lowst-lyig vibatioal-otatioal lvl of th xcitd stat, which svs as th upp las lvl. Th dcay pocss is o-adiativ ad typical liftims a i th picoscod ag. Th lasig missio occus wh th dy molcul tus to o of th vibatioal-otatioal lvls of th goud stat. As a sult, th missio spctum of a dy molcul has a boad cuv ad is omally shiftd fom th absoptio spctum.

164 45 At thshold, th coditios fo dy lasig at fqucy ω a that th gai G (ω) b qual + to th ffctiv cavity loss L. Supposig N ad N a th dsity of th xcitd ad goud L stat molculs, spctivly, ad σ (ω) ad (ω) a th iducd missio ad absoptio σ a coss sctios, spctivly, ths coditios may b witt as [34]: + G( ω ) = σ ( ω ) N σ ( ω ) N L (6-4 ) L L a L = dg dω ω L = (6-5 ) Itsity absoptio fluoscc Lasig missio Wavlgth Fig Absoptio, fluosc ad lasig of dy. Th bas idicat a thmodyamic avag ov th vibatioal sublvls of th lctoic lvls. It is covit to xpss th gai i tms of th fluoscc spctum K (ω), givig [35]: G( ω) = ( N + h[( ω μ ) / T ] N πc )( ) K ( ω) ω (6-6)

165 46 wh is th factiv idx of th host mdium, ad μ th chmical pottial diffc aisig fom th gal sult a ( ω) [( ω μ ) / T ] σ = h. I aalyzig quatio (6-6) at thshold, σ ( ω) w fid th dy lasig thshold fqucy ω L icass with cavity loss L ad dcass with + dy coctatio (N = N + N ). Fig. 6.6 dpicts th absoptio ad fluoscc spcta alog, as wll as th lasig missio. Expimtal sults [34] ag with th thotical aalysis that th dy lasig thshold + fqucy ω icass with cavity loss L ad dcass with dy coctatio (N = N + N ). L This will also b show i ou xpimtal sults. As th fluoscc spctum of th dy shifts towads log wavlgths as compad with th absoptio spctum, lasig tds to occu o th log wavlgth sid of th pa of th fluoscc spctum. Whil th fluoscc spctum pulls th lasig wavlgth clos to th fluoscc pa with low valus, th absoptio spctum tds to foc th lasig wavlgth fath away fom th fluoscc pa. Gally, fo a isotopic dy molcul, th photoxcitatio is issitiv to th polaizatio stat of th xcitatio light. Whil fo a aisotopic dy molcul, th photoxcitatio highly dpds o th polaizatio stat of th pumpig light, du to th dichoism of th dy molculs [67]. Fo a positiv dy, th photoxcitatio of th dy molculs by a lialy polaizd pumpig light souc with polaizatio paalll to th dipol momt of th dy molculs is lag tha that ppdicula to th dipol momt. If all of th dy molculs a oitd i th sam dictio, th photo-xcitatio of th sampl is polaizatio dpdt. If all th dy molculs a adomly distibutd i a sampl, thy fuctio as a isotopic mdium ad th photoxcitatio is polaizatio idpdt.

166 47 Las missio tuig withi th missio bad of th dy is accomplishd by discimiatig agaist most of th fqucis, i.., by maig th cavity loss lag tha th gai fo most fqucis. Th taditioal mthod of accomplishig this mas us of th Littow aagmt stchd i Fig. 6.7, which shows that tuig th ct pa of th las is achivd by otatig a diffactio gatig, which flcts adiatio of wavlgth λ oly i thos dictios satisfyig th Bagg coditio: dsi(θ)=mλ, m=,,3, (6-7) wh d is th spacig btw th lis of th gatig. Wavlgths ot satisfyig (6-) a ot fd bac alog th cavity axis ad cosqutly hav lag losss. Thus, th badwidth of th las adiatio is gatly ducd, ad tuig is accomplishd by otatio of th gatig. Basically, th Littow aagmt fo th wavlgth tuig of a dy las is a mchaical tuig mthod. Flashlamp θ Dy Output Diffactio Gatig Mio Fig Littow aagmt tus th ct pa of a las by otatig th gatig.

167 48 Fig. 6.8 shows th molcula stuctus of th las dys Pyomth 58 (P58) (a) ad DCM (b). Whil th dy P58 has a lasig missio wavlgth agig fom 545 m to 585 m at th xcitatio of a Nd:YAG(53) las, th wavlgth ag of lasig missio of th DCM dy covs 6 m~ 655m. W hav usd both P58 ad DCM i ou dy-dopd HPDLC systms that will b discussd lat i this chapt. H 3 C(CH ) 3 N + N B - F F (CH ) 3 CH 3 (a) N O NC CN (b) Fig Molcula stuctu of th lasig dy Pyomth 58(a) ad DCM(b).

168 Itoductio to Photoic Bad Gap Matials Photoic bad gap (PBG) matials, with cosidabl pomis fo th mgig gatio of ao- ad msoscal optolctoic compots [68, 69], hav ctly b utilizd fo high tchology applicatios. O-, two- ad th-dimsioal PBG matials hav b studid fo mo tha a dcad. Rfactiv idx modulatio at piodicitis compaabl to optical wavlgths iflucs th bhavio of photos i a ma ai to th ifluc of th cystal lattic o th bhavio of smicoductos [7]. Costuctiv ad dstuctiv itfc of a popagatio wav lads to th hacmt ad dpltio of th dsity of stats of sustaiabl optical gis (fqucis) withi a matial. A wll-dfid dsity of stats xists i ths matials ad popagatio of spcific gis o photos may b compltly, o patially pohibitd by th physical stuctu of ths photoic dvics. Complt suppssio of th dsity of optical stats, wh th popagatio of photos with a spcific gy is pohibitd, dpds o th stuctu (symmty-dfid spac goup), compositio, ad factiv idx cotast of th PBG [7]. Ths chaactistics a th basis fo xt gatio optical wavguids, sso platfoms, lass, ad display dvics. Vaious fabicatio tchiqus a big usd to fabicat PBG matials, such as advacd lithogaphic tchiqus [7,73,74], lay-by-lay chmical vapo dpositio [75], colloidal cystal gowth [76], slf-assmbly of bloc co-polyms [77,78], ad two-photo micofabicatio [79]. Syoymous with th lctoic bad gap poptis of smicoductos, photoic cystals ad PBG matials xhibit itstig poptis at o a thi bad gaps. Dowlig, Bowd ad co-wos pfomd aly wo i this fild. Thy dvlopd a thotical famwo whby lasig could b achivd i a piodic stuctu composd of matials of difft dilctic

169 5 costats [8]. Lasig has sic b achivd i a wid vaity of photoic cystals composd of vaious ogaic ad i-ogaic matials xhibitig bad gap stuctus. 6.6 Itoductio to Lasig i Liquid Cystal Matials A vaity of liquid cystal matials hav b usd fo both lasig ad amplifid spotaous missio. Th stuctu of liquid cystal matials is asily cotolld though sufac tatmt tchiqus, hologaphic mthods, additiv matial such as chial dopats, ad cofiig gomtis. PBG stuctus hav b catd i cholstic [8-86] ad folctic liquid cystals [87-88], liquid cystal lastoms [89, 9], ad polym dispsd [9, 9] ad hologaphic-polym dispsd liquid cystals (HPDLCs) [93-95]. With th xistc of a bad gap i ths stuctus, it is possibl to achiv lasig with th us of th pop las dys ad pumpig soucs. Th most familia mdium fo lasig fom liquid cystal basd photoic cystals is th cholstic liquid cystal. Cholstic liquid cystals a chial matics, wh th hadss of th costitut molculs causs th oitatio of th local matic dicto to vay lialy with positio alog th hlix axis, which is ppdicula to th dicto. Th spatial piod of th stuctu is dfid as th pitch, which is dtmid by th twistig pow ad th coctatio of th chial costituts. As a cosquc of th bifigc of th liquid cystal ad th piodicity of th hlical cholstic stuctu, light popagatio alog th hlix axis is fobidd i a ag of wavlgths, icidt light is stogly flctd wh th wavlgth of light lis i this bad ad has th sam hlicity as th cholstic. Th dgs of this flctio bad a at wavlgths of p ad p, wh ad a th odiay ad xtaodiay factiv o idics of th liquid cystal, ad p is th pitch [96]. o

170 5 Bcaus of th xistc of th slctiv bad, cholstic liquid cystals a D photoic badgap matials with a badgap stuctu, ad allow fo th possibility of lasig without th xtal mios that usually foms a las cavity. Wh a fluosct dy is dissolvd i th cholstic host, th popagatio of o omal mod of th mittd light is fobidd if th pa of th fluosct missio of th dy is i th slctiv flctio bad of th cholstic. Lasig i dy-dopd cholstics was poposd as aly as th 97s by L.S. Goldbg ad J.M. Schu [97]. Thi poposd lasig mdium compiss of a mixtu of a stogly fluosct dy, 7-dithyl-4-mthyl coumai, ad a cholstic liquid cystal solutio of 4 pct cholstyl olyl caboat, 3 pct cholstyl chloid ad 3 pct cholstyl moaoat. Th dy is pumpd with light at its absobig wavlgth of 34 m, ad mits light i a bad ctd at a wavlgth of about 45 m. Oth obsvatios of lasig i dy-dopd cholstic liquid cystals w mad appoximatly thity yas ago [98]. Yabloovitch thotically dscibd th lasig actio fom ths stuctus, ad oth photoic cystals, though th us of distibutd fdbac thoy [99]. Kopp ad co-wos hav ivstigatd lasig at th bad dg of cholstic liquid cystal matials [,]. Chaishvili ad co-wos ad hav do xtsiv wo towad catig tuabl cholstic lasig soucs i both th ulta-violt ad visibl gims [-4]. A.Muoz t al. [84] futh potd UV lasig i cholstic liquid cystals without dy matial, wh th cholstic liquid cystal act as both th activ matial ad a distibutd cavity host. Thi matial compiss th cholstic mixtu BL6 fom EM Idustis with ight-hadd hlicity ad th ight-hadd chial dopat 4-(-mthylbututyl)-4-cyaobiphyl) (CB-5). Lasig at difft wavlgths i th a UV a obsvd by chagig th atio of th

171 5 chial dopat, thus shiftig th dg of th flctio bad i th ag of 385 m 45 m, wh sampls a ud picoscod xcitatio at 355 m. Tuig of th lasig of a dy-dopd cholstic matial always cospods to tuig of th pitch of th cholstic. Taditioally, thmal tuig uss th tmpatu dpdc of th cholstic liquid cystal o th chial dopat; chagig th tmpatu chags th pitch of th cholstic ad i tu th dg of th fobidd bad, thus th lasig wavlgth ca b thmally tud. Oth difft mthods hav b xploitd to achiv th tuability of lasig i cholstic systms, such as applyig a mchaical stss [5] o a lctic fild [5]. A photo-iducd tuig of cholstic lasig was achivd by A. Chaishvili t al. [6, 7], basd o th photo-iducd tasfomatio of th chial dopat that allows th slctiv flctio bad to b shiftd. Th chial dopat ZLI-8 udgos a phototasfomatio wh iadiatd at UV wavlgths blow 3 m. This tasfomatio is a photo-fis aagmt. Th tuabl ag is up to 3 m; howv, th tuig spd is vy slow. 6.7 Itoductio to Dy-Lasig i HPDLC Hologaphic fomatio of PBG matials has abld a wid aag of dvic applicatios by lvagig xistig liquid cystal ad display tchologis. Hologaphic fomd PBG matials allow fo a simpl, apid fabicatio pocss by th xposu of a ppolym mixtu to a itfc patt. Mo ctly, atttio has b giv to lasig i hologaphic-polym dispsd liquid cystals. Buig ad cowos w th fist to obsv las missio fom a HPDLC [8, 9]. As thi fist tial fabicatio of gatigs with Coumai 485 (C485) addd to th p-polymizd syup was usuccssful bcaus of th photodgadatio of C485 ad th ovlap of th C485 fluoscc with th absoptio spctum of Ros Bgal, thy icopoatd C485 i th flctio gatig by simply washig a C485-butyl actat solutio ov a fomd

172 53 HPDLC sampl aft th moval of th top glass substat [8]. Wh photo xcitd with th tipld-output (355 m) of a Nd:YAG, th lasig missio occus at th bad dg of th flctio otch that was dsigd to b withi th fluoscc spctum of C485. Subsqutly, Buig ad cowo s wo has b followd by Matsui ad cowos [,], Luchtta ad cowos [93,], ad sval oths [94, 4]. A HPDLC is fomd by th xposu of a ppolym mixtu of liquid cystal, moom ad photoiitiato to a itfc patt, which has b discussd i pvious chapts. To cat a D gatig stuctu, two icidt bams compos a itfc patt. Th photoic bad gap aiss du to th idx of factio mismatch btw th liquid cystal ad polym lays. Th idx of factio of th liquid cystal plas is som avag of th odiay ad xtaodiay idics of th matial. This avag is mismatchd fom th idx of th polym by Δ.. Wh photos of ctai gis cout this idx mismatch, thy obsv it as a mio li bouday ad may flct off it. Ths flctios fom th photoic bad gap; spcific wavlgths of light will ot b abl to pass though th matial. Hologaphic-polym dispsd liquid cystals (HPDLCs) hav also b fabicatd ito twoad th-dimsioal stuctus [5], though th us of mo tha two bams to cat th complicatd itfc patt. Usig a complx stup of multipl bams, it is possibl to cat ay Bavais lattic stuctu as a HPDLC PBG matial [6], as wll as a wid aay of quasicystal stuctus [7,8]. Naly ay imagiabl 3D piodic stuctu ca b fabicatd usig this hologaphic tchiqu. Th HPDLC psts a ogaic PBG matial, which xhibits sval advatags, icludig low-cost, apid, simpl fabicatio tchiqus, ad disadvatags, low idx cotasts btw dilctic matials, ov iogaic PBG matials [8].

173 54 W hav ivstigatd th dy-lasig of HPDLC with difft stuctu ad difft matials, which will b discussd fom sctio Dy Lasig fom HPDLC of Difft Mods: Matials, Fabicatios ad Rsults 6.8. Lasig of sigl flctiv dy-dopd HPDLC I od to dmostat lasig actio i dy-dopd HPDLCs, a umb of HPDLC clls w poducd i flctio mod. A ppolym mixtu of liquid cystal, moom ad photoiitiato w usd. Th mixtu cosistd of 4 % of th liquid cystal BL38 ( =.799, o =.57, Δε = +6.4, Mc), 55 % moom (: Ebcyl 83 ad Ebcyl 4866, Ciba Spcialty Chmicals) ad 5% photoiitiato cotaiig 4 % Ros Bgal, % -phylglyci ad 86 % -viyl-- pyolido (Sigma-Aldich) to ssitiz th mixtu to visibl light. To this mixtu was addd.3% of las dy Pyomth 58 (,3,5,7,8-ptamthyl-,6-di--butylpyomthdifluooboat), availabl fom Excito, Ic. Th mixtu was thooughly mixd usig a sti ba fo sval hous to su th las dy compltly td th solutio homogously. Glass spacs w mployd i od to cotol th sampl thicss. Th sampls w pssd usig a balloo pss fo 5 miuts sultig i a homogous distibutio of th mixtu. Iitial sampls w fabicatd usig th itfc of two coht bams. Th agls of icidc fo th two bams w dtmid though th us of th Bagg coditio. Th pitch of th gatig was slctd such that th flctio bad, o photoic bad gap, was withi th ag of obsvd lasig fom PR58, which is 545 m 59 m. Th pitch, o piodicity, of th HPDLC, Λ is: Λ= λ w si(θ) (6-8)

174 55 wh λ w is th wavlgth of th las usd to wit th HPDLC gatig, θ is th agl of icidc fo th witig bams, is th avag idx of factio of th ppolym mixtu, i th cas of ou ppolym mixtu.55. Oc fabicatd, th PBG, o flctio bad will occu at a wavlgth λ = Λ. A agl of icidc fo th witig bams of 5 will sult i a pitch Λ = 87. m ad a flctd bad a λ = 58 m, idal fo th las dy Pyomth 58. Fig Lasig missio fom a flctio mod HPDLC (solid li) ad tasmissio spcta of th sam sampl (dottd li). A Billiat modl fqucy doubld Nd:YAG las opatig at 53 m was usd to pump ou HPDLCs. Whil th las has a ptitio at of Hz ad a maximum output of mj p puls, th las output was attuatd dow to appoximatly 6 mj p puls i ou xpimts. A las li filt was placd btw th lasig sampl ad th fib spctomt to bloc all light blow 54 m ad limiat th pump las i th masud spcta. Th solid

175 56 li i Fig. 6.9 shows a stog las li paig at m with a full width at half maximum (FWHM) of ~6.6 m. Th tasmissio spctum of th sampl is also show as a dottd li i th figu. Th thicss of th sampl is µm. It is cla that lasig occus at th dg of th bad gap, o flctio otch of th gatig. Th oth otch i th tasmissio spctum is du to th absoptio by th Ros Bgal usd i th photoiitiato, ad th absoptio of las dy. I od to switch th sampl, a xtal voltag of Hz squa wav was applid to th sampl. Th voltags masud agd fom V to 3 V. Lasig missio at V icmts is show i Fig. 6.. Th pa of lasig is locatd at m with a FWHM of 6.6 m at a voltag of V applid to th sampl; pa at 555. m with a FWHM of 8.8 m, at a voltag of V; pa at 555. m with a FWHM of 9.7 m, at V; ad pa at 555. m with a FWHM of. m, at 3 V. High voltags w ot attmptd as thy would hav shotd th HPDLC ad dstoyd th gatig. Fig. 6.. Switchig of th dy lasig missio fom a flctio mod HPDLC

176 Lasig of tasmissiv Dy-dopd HPDLC Tasmissio mod HPDLCs w mad with a small dopig coctatio of th las dy Pyomth 58. Ppolym mixtus cosistig of th matials idtical to th cip usd fo flctiv dy-dopd HPDLC w also usd fo ths gatigs, with 7.5 % Ebcyl 4866, 7.5 % Ebcyl 83, 4 % BL38 liquid cystal ad 5 % photoiitiato, all i wight pctag. Th photoiitiato cotaid 4 % Ros Bgal, % -phylglyci ad 86 % -viyl-- pyolido. Th difft mixtus w mad, ach with a difft dopig lvl of las dy:.5,. ad. %. Th mixtus w mixd with a sti ba fo mo tha hous to su homogity. Pump Las Focal Ls Cylidical Ls Dy-dopd HPDLC Lasig Em issio Fig. 6. Two ls w usd to gat th vtical li acoss th HPDLC gatig i od to icas th aa of th gai mdium big pumpd Th fabicatio las stup cosistd of two bams fom a Vdi fqucy doubld Nd:YAG las (53 m), ach with a bam pow of ~ W. Th bams w icidt o opposit sids of th sampl with a half agl btw th two bams of 46. This agl is dsigd to gat a

177 58 pitch lgth of ~369 m withi th HPDLC, cospodig to a bad gap aoud 45 m. A scod od gap xists at half this wavlgth, ~57 m. W xpct lasig to occu o th blu dg of th gap, o i th ag of 55 m 565 m. Sampls w fabicatd usig µm spacs to maitai sampl thicss. Exposus i th las stup w caid out fo miut. I th las pumpig stup, two lss w placd btw th pump las ad th lasig sampl. Th fist ls was a focal ls usd to focus th pump bam dow to a spot ad th xpad th bam, th scod was cylidical ls usd to tasfom th bam ito a vtical li acoss th HPDLC gatig, icasig th aa of th gai mdium big pumpd. Th stup is show i Fig. 6.. Lasig was masud fom th dg of th sampl, paalll to th gatig vcto of th HPDLC. A fib spctomt with a solutio of ~ m was usd fo lasig itsity masumt. Uli th flctiv HPDLC i th pvious sctio, lasig fom th dy-dopd tasmissiv HPDLC is dpdt o th polaizatio stat of pump bam. Fo th covic of discussio, h w dfi th S-compot of th pumpig las as th polaizd compot with polaizatio dictio paalll to th gatig dictio o lasig missio dictio, th P- compot with polaizatio dictio ppdicula to th gatig dictio. Masumts w fist mad of th HPDLC sampl with.5% dy coctatio. Fig. 6. shows th missio as th pump bam polaizatio is chagd fom lia s-polaizd to lia p-polaizd by otatig a λ/ wavplat 45 btw th polaiz ad th pumpig las. Th polaiz is st with its dictio paalll to th gatig dictio of th HPDLC. Th pumpig light polaizatio dpdc of th lasig missio was also obsvd fo th dy-dopd HPDLC sampls with difft coctatios (% ad %) of las dy, which a show i Fig. 6.3 ad Fig. 6.4.

178 59 Th phomo of th pumpig light polaizatio dpdc of th lasig missio ca b xplaid by th aisotopic aligmt of h dy molculs isid th HPDLC, as w hav discussd i pvious sctios. Fig. 6.5 futh illustats th micostuctu of th dy-dopd tasmissiv HPDLC systm. Whil altatig liquid cystal-ich lays ad polym-ich lay a gatd duig th polymizatio pocss ud th las itfc, th dy molculs accumulat i th liquid cystal lay much mo tha i polym lay ad a aligd alog th dictio of th liquid cystals, which, o avag, is ppdicula to th gatig dictio. Thus a p-polaizd pump las will gat much mo lasig missio. Lasig Vs Polaizatio Stat of Pumpig Light (.5% Dy Coctatio) 6 4 Lasig Itsity (Ab. Uits) Wavlgth (m) S-Polaizd P-Polaizd Fig. 6.. Lasig missio of th sampl with.5% Dy coctatio as th pump bam polaizatio is chagd fom s-polaizd to p-polaizatio.

179 6 Lasig Vs Polaizatio Stat of Pumpig Light (% Dy Coctatio) Lasig Itsity (Ab. Uits) Wavlgth (m) S-polaizd P-polaizd Fig Lasig missio of th sampl with % Dy coctatio as th pump bam polaizatio is chagd fom s-polaizd to p-polaizatio.

180 6 Lasig Vs Polaizatio Stat of Pumpig Light (% dy coctatio) Lasig Itsity (Ab. Uits) Wavlgth (m) S-polaizd P-polaizd Fig Lasig missio of th sampl with % Dy coctatio as th pump bam polaizatio is chagd fom s-polaizatio to p-polaizatio.

181 6 P-polaizd Pump Las S-polaizd Pump Las Polym lay LC lay : Dy : LC Fig Dy molculs a distibutd i th liquid cystal lays ad a aligd with th liquid cystal i th sufac, ad a ppdicula to th gatig dictio. Masumts w also mad to ivstigat th thshold atu of ths sampls, o of th y fatus of a lasig systm. Th spctum was acquid at icasig pump gis with icidt p-polaizatio, ad th pa itsity was codd fo ach spctum. Fig. 6.6 ad Fig. 6.7 show th itsitis of lasig missio icas with th pumpig gis, fo dy-dopd HPDLC sampls with dy coctatio of.5% ad %. It is appat that th FWHM of th lasig pa at aoud 55 m is ~ m, at th miimum solutio of ou spctomt. Th missio pa gows sigificatly with icasig pump pow. Wh th pumpig gy achs th high lvl, th lasig itsity is satuatd. Wh th pa itsity is plottd as a fuctio of th pump gy, a thshold ca b s i th ighbohood of 8 µj fo th sampl with a dy coctatio of.5%. Fig. 6.8 shows th missio itsity of th lasig pa at

182 63 vaious pump gis fo th sampl with a dy coctatio of.5%. Th thshold is dtmid to b ~8 µj. Lasig Emissio (.5% dy coctatio) 7 Lasig Itsity (Ab. Uits) Wavlgth ( m ) 9 Mico-J 5.5 Mico-J Mico-J 3 Mico-J 44 Mico-J 7 Mico-J Fig Lasig missio at vaious pump gis i a sampl with.5% dy coctatio.

183 64 Lasig Emissio (% dy coctatio) Lasig Itsity (Ab. Uits) Wavlgth ( m ) 3 Mico-J 44 Mico-J 66 Mico-J 77 Mico-J 9 Mico-J Mico-J 45 Mico-J Mico-J Fig Lasig missio at vaious pump gis i a sampl with % dy coctatio. Fig Pa missio itsity at vaious pump gis. A thshold is s at ~8 µj. Sampl has dy coctatio of.5%.

184 65 Lasig at Vaious Voltags Lasig Itsity (Ab. Uits) Wavlgth (m) v 5 v v v Fig Effct of lctic filds o lasig i a tasmissio mod HPDLC. Egy of Pumpig las is μj. Th lcto-optic spos was obsvd fo ths lasig systms. A lctic fild was applid acoss th sampl dopd with a. % las dy. Th applid voltag agd fom V to V. Th masud spcta at V, V ad V a show i Fig W obsv th missio falls off by a facto of two as a voltag of V is applid acoss th sampl, as compad to th zo voltag cas. Th gatig is sstially big switchd off by th lctic fild, allowig th factiv idx of th liquid cystal doplts to b idx matchd

185 66 to that of th polym. Futh wo is dd to optimiz th matials, pfctly matchig th odiay idx of th liquid cystal ad th idx of th polym Multipl-Mthod fo Lasig Tuig I od to tu th wavlgth pa of th las missio, sval mods of opatio a poposd, as show i Fig. 6.. Whil Figus 6.(a) ad (b) show th basic picipl of a HPDLC as discussd bfo, Fig. 6.(c) shows a th pal stac with a boad bad icidt whit light, Δλ W, ad th flctio bads λ B, λ B, ad λ B3, whos pa wavlgths a dictatd by Bagg s law, λ B =d<> fo omal icidt light, wh d is th pla thicss. This is a vy attactiv appoach to cat a tuabl las lmt sic th giv Bagg flctio bad ca b tud o o off lctically. Fig. 6.(d) shows th stuctu of a chipd HPDLC, wh th flctd Bagg pa, λ(x i ), is a fuctio of th spatial positio i th sampl, x. A chipd HPDLC is obtaid by catig th itfc patt with divgig bams thby spatially chagig th thicss of th Bagg plas, d, withi th sampl as a fuctio of positio. Fig. 6.() shows a sigl lctically tud HPDLC, wh th pa wavlgth, λ(v i ), ca b tud as a fuctio of applid voltag. Whil i Fig. 6. w hav alady show th lctical tuig of th lasig pa, though th tuig ag is oly ~ m, w cofiguatios ad matials a cssay to abl a lag modulatio of th factio idx of HPDLC, i od to xpad th tuabl ag Lasig Tuig i Stac of HPDLCs As a possibility fo catig a tuabl las souc, th stacd HPDLC cofiguatio was fabicatd as show i Fig. 6.(c).

186 67 Fig. 6.. Vaious mods of opatio to tu th wavlgth pa of th lasig. Lasig Switchig of HPDLC stac 35 3 Lasig Emissio (Ab. Uits) Wavlgth (m) V 5 V 5 V Fig. 6.. Stacd gatig cofiguatio fo tuabl lasig. Th gatig with th small pitch, low flctio bad is i a zo voltag stat, whil th lag pitch gatig has a fild applid acoss it to switch off lasig

187 68 Two HPDLCs dopd with difft las dys, at difft pitchs, w fabicatd spaatly usig th pocdus mtiod abov ad th sadwichd togth. O cll cotaid th las dy Pyomth 58 ad th oth was dopd with DCM; ach cll was µm thic. Ths stacd clls w th placd i fot of th pump bam ad th lasig missio was obsvd with th spctomt. Two las missio lis w obsvd. Th solid li i Fig. 6. shows th lasig missio wh o voltag was applid. A fild of icasig stgth was applid acoss o of th clls as th oth was lft i a zo fild stat, as show by th dashd ad dottd lis i Fig. 6.. Tasmissio of HPDLC Gatigs Tasmissio (Ab. Uits) Wavlgth (m) Gatig A Gatig B Fig. 6.. Tasmissio of th two gatigs usd i th stac. Gatig A is dopd with dy P58, ad gatig B is dopd with dy DCM.

188 69 Th spctal spos of th two gatigs of th stac w masud spaatly, as show i Fig. 6.. Gatig A is dopd with dy P58, ad gatig B is dopd with dy DCM. Compaig Fig.s 6. ad 6., it is claly show that th wavlgth of th lasig missio of th lft pa (i Fig. 6.) is locatd at th lft dg of th flctio bad i gatig A, ad th wavlgth of lasig missio of th ight pa (i Fig. 6.) is locatd at th lft dg of th flctio bad i gatig B. W hav dmostatd that th catio of stacd, dy-dopd HPDLCs is a viabl optio fo multipl li lasig dvics Lasig Tuig i Chipd HPDLC Chipd HPDLCs with flctio bads agig btw 56 m ad 59 m w fabicatd. Th followig fomulatio has b usd fo sampl ppaatio: 55% homogous ppolym mixtu composd of a photo-polymizabl utha acylat (fom Ciba Spcialty Chmicals), 45% liquid cystal (BL38 fom EM Idustis, =.799, o =.57, Δε=+6.4) ad 5% photoiitiato (Ros Bgal, -phylglyci, ad -viyl--pyolido; all availabl fom Sigma-Aldich). Th mixtu was placd btw two AR-ITO coatd glass substats; glass spacs w mployd i od to cotol th sampl thicss ad th sampls w pssd usig a balloo pss fo 5 miuts sultig i a homogous distibutio of th mixtu. Two additioal lss w usd i th hologaphic witig systm to gat a chipd gatig. Ths lss, placd i th path of th split bams, catd divgig bams that catd th spatially vayig gatig spacig withi th sampl. Wh tstd with a spctomt, a shift i th locatio of th flctio otch of th tasmissio spcta was obsvd wh th sampl was pobd alog o dictio. Sampls with shifts as high as m w fabicatd. Mo divgt bams ad lag sampls ca b usd to cat lag shifts of th lasig pa.

189 7 Two lss with focal lgths of mm w usd i ou stup to cat divgig bams at th cll. Th sampls a xposd to a 53 m Vdi las fo 3 scods at a pow of W. A Oca Optics fib spctomt was usd to masu th tasmissio of th fomd gatig withi th sampls. A flctio otch vayig as a fuctio of positio i th sampl lativ to th spctomt was obsvd i all of th sampls mad with a wavlgth ag btw 56 m ad 59 m. Wh masud at two opposit dgs of th sampl, a chipd HPDLCs has flctio otchs with miima i th tasmissio spcta btw 56 m ad 57 m. This shift, at th spots i th tasmissio spcta, is s i th top of Fig Ths sampls w pumpd as dscibd abov; th lasig spcta w masud at th sam th spots ad th sultig spcta a show i th bottom of Fig Las missio appas at th dg of th bad gap, o flctio bad, fo ach poit. Th lasig pas a locatd at 554.4, 555., ad m fo th lft, middl ad ight poits, spctivly. This cospods to appoximatly a 4 m shift i th lasig wavlgth. It is cla ths gatig stuctus hav spatial tuability. I additio to this spatial tuability, lasig fom ach spot of ths chipd dy-dopd HPDLCs a also switchabl, as is s i Fig Th filds cssay fo switchig a chipd HPDLC a compaabl to thos quid fo switchig a o-chipd cofiguatio. Th pa of las missio is at m with a FWHM of 4.9 m, at V; th pa of las missio is at m with a FWHM of 5. m, at V; th pa of las missio is at m with a FWHM of 7. m, at V,; th pa of las missio is at m with a FWHM of 7.4 m, at 3 V.

190 7 Fig Tuig of a chipd HPDLC. Tasmissio spcta at lft (solid), middl (dashd) ad ight (dottd) poits o th sampl (top); ad lasig missio at lft (solid), middl (dashd) ad ight (dottd) poits o th sampl (bottom). Fig Switchig of a flctio mod chipd HPDLC

191 7 6.9 Two Dimsioal Dy-Dopd HPDLC Lasig D stuctus i PDLC matials ca b fabicatd though multipl-bam hologaphy. Fo N lia polaizd pla wav bams, th iadiac of th itfc patt ca b xpssd as: ] ) ( xp[ ) ( * i E E I i j N i i N j j v = = = (6-9) If th cipocal wav vctos i j ji G = fom a cipocal lattic, th th ) ( I has a piod stuctu with th scal of lattic compaabl to th wavlgth of light. Th ability to maipulat th wav vctos ad polaizatio dictios allows fo th dsig of D o v 3D itfc patts [9-]. W will discuss a 4-bam itfc i dtails. Th 4 bams a all of th sam pola agl θ, vly distibutd at Δφ = 9 itvals aoud th azimuth, ad hav th sam itsity. Fo simplicity, lt us assum that bams li i th X-Z pla ad hav th polaizatio dictios alog th Y dictio, whil th oth bams li i th Y-Z pla ad hav a polaizatio dictios alog th X dictio. Th 4 bams ca b aalytically xpssd as: ] [ ) ( ) ( t i z x i t i z x i y E E z x z x ω ω =, ] [ ) ( ) ( t i z x i t i z x i y E E z x z x ω ω =, ] [ ) ( ) ( 3 t i z y i t i z y i x E E z y z y ω ω =, ] [ ) ( ) ( 3 t i z y i t i z y i x E E z y z y ω ω = (6-) wh x ad a two uit vcto alog X-axis ad Y-axis dictios. Wh th 4 bams mt, th total lctic fild is: y

192 73 = = 4 i E i E (6-) ad ) ( E E E E E E E E E E i i = = = (6-) H, som tms, such as 3 E E ad 4 E E, disappa bcaus th polaizatio dictios a ppdicula, o. Th total itsity is: = y x v v E E E E E E E E E I = = (6-3) As 4 3 I E E E E E = = = = = ad ] [ ] [ ) ( ) ( ) ( ) ( t i z x i t i z x i t i z x i t i z x i E E E z x z x z x z x ω ω ω ω = ) cos( 4 ] ) x E E x t i z i x i x i t i z i z x x z = = ω ω (6-4) Similaly, ) cos( y E E E y = (6-5) Thus th total itsity is: )] cos( ) [cos( 4 4 y x I I I y x + + = (6-6) Fig. 6.5(a) shows th stuctu of a 4-bam itfc. As th pola agl of th 4 bams a th sam, w hav ) si( θ λ π = = y x, thus (6.6) gats a -dimtioal squalattic with itsity modulatio. Fig. 6.5(b) shows th sultig itfc patt as

193 74 dscibd by (6.6). Th lattic stuctu was cofimd by a SEM imag pviously ta by th Display ad Photoics Laboatoy of Bow Uivsity, as show i Fig Th dsigd lattic piod was idally m. X Y Z (a) (b) Fig (a) Stup fo catig 4-bam itfc patt ad (b) th sultig itfc patt; th bight (da) gios pst aas of high (low) itsity. Fig SEM imag of a HPDLC lattic gatd by 4-bam itfc.

194 75 Aoth D cofiguatio mad us of 6 icidt bams vly distibutd at Δφ = 6 icmts aoud th azimuth, ach with th sam pola agl ad sam itsity. Th 6-bam itfc sults i a D hxagoal lattic stuctu. Th bam stup ad th sultig itfc patt a show i Fig Fig (a) Stup fo catig a 6-bam itfc patt ad (b) th sultig itfc patt; th bight (da) gios pst aas of high (low) itsity. Whil ths two cofiguatios may ot xhibit complt bad gaps, bad gaps do xist alog ctai dictios, alog th uit vctos compisig th lattic, fo istac; fo this aso a stog lasig itsity should b obsvd alog ths dictios wh lasig dy matials a dopd ito th HPDLC. W fabicatd both 4- ad 6-bam D dy-dopd HPDLC gatig stuctus usig th optical stup discussd abov. Th pola agls of icidc fom th omal of th glass substat w θ = fo th 4- bams stup. Fom 6.9, w ow th pitch of th 4-bam stup is dcidd by

195 76 π π Λ= = = x y λ =53 m, fo th witig las wavlgth 53 m ad pola agl si( θ ) θ=. y x Fig (a) Isoitsity plot fo fou-bam fabicatio with dictios of th bad gap ad subsqutly lasig; ad (b) lasig fom this stuctu dopd with th las dys Pyomth 58 (solid li) ad DCM (dottd li). Lasig missio is masud alog x-dictio. Th matials usd i th fabicatio of th -D dy-dopd HPDLC a basically th sam as th mixtus i fabicatig th -D sampls. Th ppolym mixtu cosistd of 4 % of th liquid cystal BL38 ( =.799, o =.57, Δε = +6.4, Mc), 55 % moom (: Ebcyl 83 ad Ebcyl 4866, Ciba Spcialty Chmicals) ad 5% photoiitiato cotaiig Ros Bgal, -phylglyci ad -viyl--pyolido (Sigma-Aldich) to ssitiz th mixtu to visibl light. To this mixtu was addd.3% of o of th las dys Pyomth 58 (,3,5,7,8-ptamthyl-,6-di--butylpyomth-difluooboat) o DCM (4- Dicyaomthyl--mthyl-6-p-dithylamiostyyl-4-H-pya), both availabl fom Excito,

196 77 Ic. Th mixtu was thooughly mixd usig a sti ba fo sval hous to su th las dy compltly td th solutio homogously. I th 4-bam stuctu, a bad gap is xpctd i x-dictio, y-dictio ad diagoal dictio, as show i Fig. 6.8(a). Wh viwd alog ths dictios, th coss sctio of th HPDLC sampl will appa as a simpl D gatig stuctu, li that of th D gatigs fabicatd pviously. Wh masud fom th x-dictio (ad th sam fo y-dictio), las missio pad at 558 m with a FWHM of 5 m (solid li) was obsvd i th sampl dopd with dy Pyomth 58; lasig missio pad at 64.3 m with a FWHM of. m (dashd li) i th sampl dopd with th dy DCM. Both of th lasig missio sults a show i Fig. 6.8(b). Fig (a) Isoitsity plot fo six-bam fabicatio with dictios of bad gap ad subsqutly lasig; ad (b) lasig fom this stuctu dopd with th las dys Pyomth 58 (solid li) ad DCM (dottd li).

197 78 W also fabicatd a 6-bam D dy-dopd HPDLC gatig stuctu usig th optical stup discussd i Fig. 6.7(a). Th pola agls of icidc fom th omal of th glass substat w θ = 7. Th 6-bam stuctu cotais th axs alog which lasig is xpctd, as s i Fig. 6.9(a). Las missio of th sampl dopd with PM58 pad at m, with a FWHM of 5. m (solid li); Lasig of sampl dopd with DCM pad at 63 m, with a FWHM of 4.8 m (dashd li), both a show i Fig. 6.9(b). Compaabl lasig was obsvd alog all th axs. 6. Lasig of Polaizatio Gatig I most applicatios of hologaphy, th mas of itfc is th itsity itfc, as most hologaphic matials a ssitiv to xposu itsity. Rctly a fw matials hav b foud to b ssitiv to th local polaizatio dictio ad itsity, ad thy hav b utilizd to cod th so-calld polaizatio itfc. Th fist dmostatio of ta itfc patt codd usig two othogoally polaizd bams was poposd by Kalichashvili [3], ad futh cofiguatios w systmatically studid by Niolova ad Todoov [4]. Th itfc of two othogoally polaizd bams givs a uifom itsity, howv, th distibutio of th polaizatio dictio is spatially modulatd. Two spcial ad itstig cass a cosidd as show i Fig. 6.3: () Two bams a lialy polaizd with othogoal polaizatio dictios (Fig. 6.3(a)); () Two bams a ciculaly polaizd with opposit haddss, o o lft-hadd ad th oth ight hadd. (Fig. 6.3(b))

198 79 Fo th two ciculaly polaizd bams with difft haddss, thy ca b dscibd i th Jos matix mthod by: E =, ad E. Assumig th is o diffc iπ / = iπ / of optical path lgth wh th two bams mt at th poit x =, th th diffc of th optical path (OPD) alog th x-axis is dcidd by: OPD = xsi( θ ), ad th phas mismatch Btw th two bams is 4π xsi( θ ) Δ =, thus th total lctic fild is giv by: λ E = iδ / iδ ( cos( Δ / ) cos( / ) iδ / + = = iπ / iπ / iδ / cos( π / Δ / ) Δ si( Δ / ) H it is itstig that th polaizatio stats alog th x-axis a all lialy polaizd, with piodic modulatio of th polaizatio dictios. Bcaus of th fact that th polaizatio a a π λ stats ad a th sam, th piod alog x axis is Λ = =. b b Δ / si( θ ) Similaly, cosidig th two lialy polaizd bams with qual itsity, th total lctic fild of itfc is giv by: E = E + E iδ Thus, th polaizatio is also modulatd alog th x-dictio. As th mophology of th liquid cystal systm is asily cotolld usig sufac aligmt ad hologaphic tchiqus, liquid cystal ad polym ca b utilizd fo polaizatio codigs of polaizatio gatig. Th optically iducd molcula oitatio i a matic LC film has b wll studid both thotically ad xpimtally. A lia photopolymizabl polym (LPP) was potd by Cawfod s goup at Bow Uivsity to cat a polaizatio gatig o a sufac aligmt lay fo a matic liquid cystal, which popagatd ito th bul matic

199 8 [6,7]. Fig. 6.3 shows two micoscop imags of a cll of a polaizatio gatig. Whil o substat of th cll is coatd with th polyimid SE-75 to gat homotopic aligmt, th oth substat is coatd with th LPP matial fom Rolic ad is xposd to two itfig las bams of cicula polaizatio with opposit haddss. Th piod of th gatig is 7.5 μm. I Fig. 6.3(a), th imag is ta with th cll btw two cossd polaizs wh o voltag is applid. Th piodic stuctu of th liquid cystal aligmt dictio domiatd by th substat with th LPP matial coatig is claly show i th micoscop imag. Wh a high voltag is applid, th liquid cystals chag to a homotopic aligmt ad a blac imag is show i Fig. 6.3(b). Th diffcs btw a HPDLC gatig ad a polaizatio gatig a that: () HPDLC gatig is composd of two difft matials (liquid cystal ad polym) i a piodic stuctu, whil a polaizatio gatig is composd of pu liquid cystal with piodic modulatio of th dicto aligmt; () I fabicatio, a HPDLC gatig dos ot d sufac aligmt lay, whil a polaizatio gatig ds th LPP matial fo th liquid cystal aligmt ad fo codig th piodic modulatio of th polaizatio causd by th itfc of th two hologaphic witig bams. Th polaizatio-modulatd gatig ca also b tatd as a photoic bad gap (PBG) stuctu. Wh a lasig dy is dopd ito th systm, lasig missio is xpctd. I od to obtai idal coditios fo a distibutd fdbac missio, th piodicity, o pitch of th polaizatio gatig, is slctd so that a dg of th flctio bad gap would li withi th fluoscc bad of th las dy. Accodig to Koglic s coupld wav thoy [8,9], th wavlgth of th hacd missio fom this Bagg gatig stuctu is dcidd by: Λ = λ,

200 8 wh is a ffctiv factiv idx, Λ is th pitch of th piodic mdium ad is a itg pstig th diffactio od of th gatig. W fabicatd th dy-dopd matic liquid cystal sampls with a polaizatio gatig stuctu as discussd. Th LPP matial (ROLIC) was spi-coatd oto two glass substats with a ati-flctio coatig o o sid, a thi lay of idx-matchd idium ti oxid (ITO) coatig o th oth. 5 µm fib spacs w spad o o substat ad th th uifom mpty clls w assmbld with poxy. Ths sampls w th xposd to a itfc patt gatd by two ciculaly polaizd bams of idtical itsity ad opposit haddss, fom a A + las opatig at 35 m. Each bam cotaid a pow of appoximatly 4 mw. Th xposu tim was about 3 scods. Ud th ultaviolt adiatio of ths two itfig bams, th LPP molculs polymiz with thi molcula axs paalll to th local polaizatio dictio that was piodically modulatd as show i Fig. 6.3(b). Th matic liquid cystal BL38 ( =.7999, o =.57, Δε = +6.4, EM Idustis) was chos, fo its lag bifigc ad dilctic aisotopy, as th host solvt fo a ogaic las dy Pyomth 58 (PM58, Excito, Ic.). Th coctatio of th dy was ~.5 wt.- %. Th dy-dopd liquid cystal mixtu was mixd fo appoximatly o hou usig a sti ba to su a homogous mixtu, bfo big capillay filld ito th xposd clls. Ths sampls w pumpd usig a 53 m fqucy doubld Nd:YAG las (Quatl) with puls widths of 9 s, a ptitio at of Hz ad a vaiabl bam itsity. Th icidt agl of pumpig bam o th sampl is st at appoximatly 45. Th missio fom th sampl was masud fom th dg of th gatig, as show i Fig. 6.3(b). Two lss w usd to focus th icidt bam o th gatig stuctu; a plao-covx ls focuss th icidt bam to a poit ad a cylidical ls spads th bam alog a dictio paalll to th gatig vcto of

201 8 th stuctu. This stup icass th aa of gio xcitd by th pump bam withi th gatig stuctu, as is show i Fig. 6.. Las missio fom ths sampls was masud ad is show i Fig Th lasig missio pa has a full width half maximum (FWHM) of ~5 m, compad with that of th FWHM of th fluoscc bad (>5 m). (a) y E z θ E x (b) y x E z θ E x Fig (a) Two lialy polaizd bams with othogoal polaizatio dictios; (b)two ciculaly polaizd bams with opposit ss of clocwis. [7]. x

202 83 (a) (b) Fig Micoscop imags of a cll of polaizatio gatig btw two cossd polaizs. (a) o voltag is applid; (b) V voltag is applid. Fig Witig bam ad pump bam fo fabicatio ad lasig missio tstig of th polaizatio gatigs.

203 84 Fig Lasig missio fom a liquid cystal polaizatio hologaphy gatig. Fig Thshold of las missio fo th dy-dop liquid cystal polaizatio gatig.

204 85 Th thshold of ths sampls was masud ad is show i Fig Th thshold pump gy is ~ 5 µj. Blow this thshold poit, th lasig missio gy chags slowly lativ to th at of chag of pump gy. Abov this thshold poit, th at of chag icass mo dastically. Th FWHM of missio dops dow to ~5 m ud a 7 µj pump bam. Also masud was th pump bam polaizatio dpdc of ths gatig stuctus. A half-wav plat was placd aft th polaiz i fot of th pumpig las. Th otatio of th polaizatio was cotolld by th otatio of a half wav plat. A 5% icas i lasig missio was obsvd, as show i Fig. 6.35, wh th icidt polaizatio stat was otatd fom s- polaizatio to p-polaizatio. As w hav discussd i th lasig of tasmissiv HPDLC gatigs, dy molculs td to alig thmslvs alog with th liquid cystals. I th stuctu of th polaizatio gatig, th sufac aligmt dictios of th liquid cystal otat withi th pla of th substats alog with th modulatio of th polaizatio dictios. Wh th pump las is at omal icidc, both th p-polaizd bam ad th s-polaizd bam a idtical to th dy molculs that hav a uifom siusoidal distibutio i difft i-pla dictios. Howv, wh th icidt agl of th pump bam is 45, th lctic fild of th p-polaizd light still lis i pla, whil that of s-polaizd light has oly 5% that lis i pla, th oth 5% has th lctic fild ppdicula to th pla of th liquid cystals ad th dy molculs ma littl cotibutio to th lasig missio.

205 86 Fig Lasig missio icass by 5% as th icidt polaizatio is otatd fom s- to p-polaizatio. Th switchig ad tuig of th lasig of liquid cystal polaizatio gatigs was also masud. Wh a lctic fild was applid to a sampl pumpd with a p-polaizatio las, alog with th switchig off of th polaizatio gatig, th itsity of th lasig missio dops by 6%, alog with a d shift of th wavlgth of missio by appoximatly 5 m, as is s i Fig Th applicatio of a lctic fild applid to a sampl pumpd by s-polaizd light had o ffct o th output itsity of missio, but did d shift th lasig wavlgth by 7 m.

206 87 Fig Effct of a applid lctic fild o a liquid cystal polaizatio gatig pumpd by p-polaizd light.